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WOODS HOLE
OCEANOGRAPHIC INSTITUTION

LABORATORY
BOOK COLLECTION

WOODS HOLE OCEANGGKAPHIC Il^STITUTION

REZ'-'SaMCE LIBRARY .

PHYSICAL OCEANOGRAPHY

Volume I

PHYSICAL
OCEANOGRAPHY

ALBERT DEFANT

Dr. phi/.. Dr. rer.nat., h.c.

EMERITUS PROFESSOR OF METEOROLOGY AND GEOPHYSICS

at the

UNIVERSITY OF INNSBRUCK

HONORARY PROFESSOR OF OCEANOGRAPHY

at the

UNIVERSITY OF HAMBURG

and at the
FREE UNIVERSITY. BERLIN

PERGAMON PRESS

NEW YORK • OXFORD • LONDON • PARIS

1961

PERGAMON PRESS INC.

122 East 55th Street, New York 22, N. Y.
P.O. Box 47715, Los Angeles, California

PERGAMON PRESS LTD.

Headington Hill Hall, Oxford

4 &5 Fitzroy Square, London W.l.

PERGAMON PRESS S.A.R.L.
24 Rue des Ecoles, Paris Ve

PERGAMON PRESS G.m.b.H
Kaiserstrasse 75, Frankfurt am Main

1961

Pergamon Press Ltd.

LIBRARY OF CONGRESS CARD NUMBER 59-6845

Printed in Great Britain by Page Bros. {Norwich) Ltd.

Contents

PART I

Page

Preface ix

Introduction xiii

I. The Ocean 1

A. The horizontal extent and the structure of the ocean 1

1. Introduction, vertical structure of the total Earth 1

2. The horizontal extent of the ocean and its boundaries 1

3. Sea level and its variations. Chart datum 5

B. The three-dimensional structure of the ocean 10

1. Methods of recording deep-sea data 10

2. The general morphology of the sea bottom 10

3. Special characteristics of sea-bottom topography 18

4. Arrangement of the general bottom topography of the individual

oceans 27

II. The Sea-water and its Physical and Chemical Properties 32

1 . Collecting oceanographic samples 32

2. Temperature determination for all layers of the ocean 34

3. Salinity and its determination 36

4. The density of sea-water and its dependence on temperature, salinity

and pressure 41

5. Vapour pressure, freezmg point, boiling point and osmotic pressure of

sea-water 44

6. Other physical properties of sea water 48

7. The optical properties of sea water 51

8. The chemistry of the sea 64

III. Temperature in the Ocean , The Three-dimensional Temperature distribu- 8 8

TION AND ITS VARIATION IN TiME 88

1 . Heat sources, heat exchange and heat budget in the ocean 88

2. Heat transport in the sea : absorption, conduction, thermo-haline and

dynamic convection (turbulence) 94

3. Diurnal and annual variation of the temperature in the ocean 109

4. The vertical distribution of temperature in the ocean 1 1 7

5. Temperature distribution in horizontal and vertical sections 140

6. Mean vertically integrated temperature for individual oceans in zonal

rings

153

IV. The Salinity of the Ocean, its Variation in Oceanic Space and in Time 1 54

1 . Periodic and aperiodic variations of salinity 1 54

2. The horizontal distribution of surface salinity 161

3. The vertical distribution of salinity (in vertical profiles and sections) 165

4. The horizontal distribution of salinity at particular depths 179

5. Salinity in adjacent seas and sea straits 181

vi Contents

Page
V. The Density of Water Masses in the Ocean, Vertical and Horizontal

Density Distribution and its Stability 185

1 . Diurnal and annual variations at the surface 1 85

2. Density distribution at the surface of the ocean 1 87

3. Vertical density distribution and horizontal charts for different depths 1 87

4. Potential density and isentropic analysis 192

5. The vertical equilibrium in the ocean and stabiUty 195

6. The distribution of stability in the Atlantic Ocean 198

VI. The [TIS] -relationship and its Connection w^th Mixing Processes and

Large Water Masses 202

1 . Temperature as a function of salinity and large water masses 202

2. Practical significance of the [r5']-curve 203

3. The [r^S] -curve and the mixing of water masses 204

4. Further examples of the [TIS] -relationship 210

5. The water masses of the oceans 216

VJI. Evaporation from the Surface of the Sea and the Water Budget of the

Earth 219

1. Introduction 219

2. Direct measurement of the evaporation on board ship and methods

for obtaining corresponding values for the sea surface 220

3. Meridional distribution of evaporation over the whole ocean and its

determination from energy considerations 223

4. Geophysical aspects of evaporation problem 226

5. The water budget of the earth 231

6. Energy budget between ocean and atmosphere for different oceans

and oceanic regions 236

VIII. Ice in the Sea 243

1 . Formation and termmology of sea ice 243

2. Physical and chemical properties of sea ice 245

3. Ice conditions and their seasonal and aperiodic variations in Artie

and Antartic regions 257

4. Land ice in the sea 271

5. Effect of Polar- ice conditions on the atmospheric and oceanic cir-

culation 279

Bibliography 285

PART II

DYNAMICAL OCEANOGRAPHY

Introduction 299

IX. The Geophysical Structure of the Sea 301

1. Introduction 301

2. The distribution of gravity and gravity potential 301

3. The field of mass 303

4. The pressure field and its relationship to the mass fields; solenoids 304

5. The dynamical method of preparation of oceanographic data 309

Contents vii

Page

Forces and their Relationship to the Structure of the Ocean 312

1. External, internal and secondary forces 312

2. The basic hydrodynamic equations 320

3. The continuity equation and the boundary-surface conditions 323

4. Potential flow, the Bernoulli equation, impulse and the impulse form

of the hydrodynamic equations 325

5. Circulation and vorticity 329

XI. The Ocean at Rest (Statics of the Ocean) 337

1. The basic static equation and the conditions for static equilibrium 337

2. Quasi-static equilibrium and its importance in the dynamic evalu-

ation of oceanographic observations 338

3. Disturbances and re-establishment of static equilibrium 339

XII. The Representation of Oceanic Movements and Kinematics 342

1 . Methods of observation and measurement of oceanographic currents 342

2. The current field and its representation 356

3. Special cases of current fields near land and at the boundaries of water

masses (compensation currents) 370

4. Divergence of the current field and the continuity equation 374

5. The Knudsen relations 379

XIII. General Theory of Ocean Currents in a Homogeneous Sea 382

1. Introduction 382

2. Steady currents in a homogeneous sea without friction 383

3. Eddy viscosity (turbulent friction) in ocean currents 387

4. Steady currents in a homogeneous ocean under the action of external

forces 398

5. Ice drift 436

6. Inertia currents 441

XIV. Water Bodies and Stationary Current Conditions at Boundary Surfaces 45 1

1. Water bodies and the boundary surface between them 451

2. Stable discontinuity surfaces 453

3. Stable stratification of water masses 458

4. Up-and down-gliding surfaces : pulsations of stationary vortices 469

XV. Ocean Currents in a Non-homogeneous Ocean 476

1. Introduction 476

2. Relationships between current and density fields in a horizontal plane.

The law of parallel fields 476

3. Horizontal steady currents in a stratified ocean ' 479

4. Ekman's theory of density currents including friction 482

5. Oceanographic applications of Bjerknes's circulation theorem 486

6. The "reference-level" for the conversion of the relative topography of

the pressure surfaces into the absolute one 492

7. Remarks about the observational material necessary for a dynamic

computation and critical discussion of the procedure 504

8. The determination of water transport in density currents 508

viii Contents

Page

XVI. Currents in a Strait 513

1. Water stratification and water movements in sea straits 513

2. Theory of currents in sea straits 517

3. Ocean currents in individual sea straits 523

4. External influences (bottom topography, tides) on the oceanographic

conditions in sea straits 534

5. Processes in estuaries (river mouths) 538

XVII. Effect of Wind on the Mass Field and on the Density Current 544

1 . A limited and stratified sea 544

2. General conditions in the open ocean 547

3. General relationships between wind and currents 550

4. Velocity computations of oceanic surface currents in the equatorial

regions from wind data 552

XVIII. Basic Principles of the General Oceanic Circulation 556

1. Introduction 556

2. Oceanic sea surface currents 557

3. Currents caused by excess of precipitation and run-off" over evaporation 572

4. The thermo-haline circulation 574

5. Wind effects and the current system in a hydographic circular vortex 576

6. The influence of meridionally oriented coasts on the oceanic circulation 579

XIX. The Tropospheric Circulation 592

1 . The position and structure of the oceanic troposphere 592

2. The tropospheric circulation of the tropical and subtropical oceans 594

3. Other currents of the oceanic troposphere 606

4. Upwelling phenomena 643

5 . Processes at the polar boimdary of t he subtropical convergence region 656

XX. The Stratospheric Circulation 661

1. Introduction 661

2. Polar currents of the northern hemisphere 662

3. The processes which occur at the Antarctic convergence zone 669

4. Dynamics of the Antarctic circumpolar current 673

5. The Sub-Antarctic intermediate current 675

6. The Polar bottom current 680

7. The deep currents in the middle part of the oceanic stratosphere of

individual oceans 683

8. A survey of the water transports in the individual layers of the

Atlantic Ocean 688

9. TTie effect of the subtropical adjacent sea on the deep sea circulation 690

XXI. The Main Features of the General Oceanic Circulation and their Phy-
sical Exploration 694

1. The oceanic circulation in the Atlantic 694

2. Summary of present individual theories and the prospects of a com-

prehensive theory of the general circulation including the deep

layers 696

3. Model experiments on stationary planetary flow patterns 701

4. The transient response of an ocean to a variable wind stress 702

Bibliography 708

Author Index 721

Subject Index 725

{

Preface

Oceanography, the science of the ocean, has undergone a rather rapid development
during the last decades tending from a more descriptive science towards one working
according to exact mathematic-physical principles as appUed in the natural sciences.
Oceanography can be subdivided into two fundamentally different parts: (1) The
"biology of the oceans" and (2) the "physical oceanography". For the first, physical
oceanography can be looked upon as the scientific foundation, since the biology of
the oceans dealing with conditions and forms of life of all the Uving beings existing in
the oceans requires an exact knowledge of the environmental medium for these beings.
Physical oceanography in itself is a subpart of geophysical science. This book involves
physical oceanography only, the scientific progress of which has been especially fast
during the last 50 years owing to technical improvement of the working methods used
on oceanographic research vessels and also to the extensive widening of our physical
and chemical views about the phenomena occurring in the sea.

The start of the manuscript work of this book goes rather far back, to the time
when the scientific results of the German Atlantic Expedition on the research vessel
Meteor 1925-1927 were almost completed. However, these first compilations took a
considerable time and served as the basis of extensive oceanographic lectures at the
Institute and Museum of Oceanography (Meereskunde) at the University of Berhn
(1925-1945), assembled together in book form. The book was completed in its first
form at the end of the Second World War (1945). Of course, at the time, it was im-
possible to achieve a publication of the work. Consequently the first manuscript has
been rearranged several times and has, on these occasions, been revised rather extensive-
ly and completed according to the momentary state of oceanographic research. From
one point of view this circumstance may be looked upon as an advantage for the
presentation, but from the other as a drawback for the internal uniformity of the book,
since it was unavoidable sometimes to present some subjects shorter and others longer
than needed. However, a compromise was always tried and found.

More recently (1957), after some failures to achieve pubhcation of the book, two
institutions took interest : On the one hand the Deutsche Forschungsgemeinschaft in
Bonn, Germany, proposed a generous fund for a publishing house, Dietrich-Reimer,
Adrews & Steiner, Berhn. On the other hand the Woods Hole Oceanographic Institu-
tion, U.S.A. which by way of the Office of Naval Research, U.S. Navy, arranged with
the Pergamon Press, Oxford (Capt. I. R. Maxwell) the pubhcation of the book in the
Enghsh language. There were many reasons favouring a pubhcation in the English
language. Certainly international oceanographic science was hoped to be better
served because of the larger audience possible.

It was doubtful, besides, if the large funds necessary for publication could have
been raised from the German side.

These circumstances required a translation of the German text into Enghsh. Before
approaching this large task, the work had again to be revised completely and brought

X Preface

up-to-date. This time-taking job was done at the International Institute of Meteorology
in Stockholm, thanks to a kind invitation by the late Prof. C. G. Rossby shortly before
his untimely death. It remains a pleasure and self-evident duty to express my gratitude
towards the present director of the Institute, Docent Dr. B. Bolin, as well as towards
his closer co-workers, for their interest, and for all the Institute facihties at my disposal,
whereby my work benefited greatly. During my six-month stay in Stockholm the first
volume of the book was translated (Physics of the Ocean, Statics and Dynamics of
Ocean Currents) (translator Ing. H. E. Knoll, Stockholm), while the second volume
(Waves, Tides and Related Phenomena) already drafted years ago was translated by
Dr. Louis Lek, La Jolla, California. I express my thanks to both translators for the
trouble they undertook. As modes of expression differ among languages it is natural
that the detailed refinements of the originally German formulation and presentation
naturally suffered to some extent, but I hope that a still representative version of the
contents has satisfactorily been achieved. All this, however, could not have been
completed had not my son Prof. Fr. Defant, on leave from the weather service and the
University of Innsbruck, Austria, been present at the Stockholm Institute, engaged in
investigations on the General Circulation of the Atmosphere. He also devoted his time
to my work, especially concerning detailed revisions of the translations and the com-
pletion of the large amount of illustrations. For this troublesome work, which for him
also meant loss of time, I am especially grateful.

The printing of this textbook would have been doubtful had not the Office of Naval
Research, in the first place (Dr. Atkins and Mr. G. Lill, Office of Naval Research, Wash.
D.C.), generously sponsored the undertaking, at the same time conceding to my
wishes with regard to its publication. Furthermore, I wish to thank Commander
C. Palmer of the U.S. Navy, at present with the International Institute, Stockholm,
for additional help.

The publishing has been done by Pergamon Press, Oxford, in its well-known and
outstanding manner, and I express my gratitude in the first place to the publisher,
Captain I. R. Maxwell. Also to Mr. Buchanan my heartful thanks and appreciation for
the excellent drafting of the numerous illustrations.

Physical Oceanography consists of two volumes, each having two sub-parts. The
first part of Volume I deals with the spatial, material and energetical characteristics of
the water envelope of the earth, as well as with the evaporation problem and the ice in
the sea. For this reason it specially involves the physical and chemical properties of
sea-water, the spatial distributions of the oceanographic elements in the total oceanic
space and its periodic as well as aperiodic changes. The second part of Volume I
concerns the various modes of motion of sea-water in the form of ocean-currents
(dynamic oceanography). Finally, Volume II is devoted to periodic movements of the
water masses (waves, tides and related phenomena). The individual problems of
physical oceanography are discussed in as much detail and supported as far as possible
by appropriate examples and references to existing compilations of observational
data. The scientific progress of the last decades has been considered almost completely,
not only with regard to the observational facts, but also concerning the theoretical
treatment and explanation of the observed phenomena. The oceanographic literature
has been considered in its entirety to the end of May 1957. Extensive reference lists are

Preface xi

provided at the end of each sub-chapter concerning the hterature sources used and can
be considered as unique in their completeness.

A presentation of instruments and apparatus in use in oceanographic research, their
technical function and instrumental theory, was not intended to be included in the
textbook. Since the different nations engaged in oceanographic research mostly use
their own instruments and apparatus, it would be rather difficult in the frame of such a
textbook to deal with all instruments and explain their function. I beheved this to be
unnecessary since much has already been summarized by authoritative institutions and
also because a detailed textbook on oceanographic instrumentation has been com-
missioned from the international side.

The contents of the book formed the basis, as already mentioned, for my lectures on
physical oceanography held at the University of Berlin Institute and Museum for
Oceanography (1927-1942); later on, until 1953, they were the basis for my lectures
held at the University of Innsbruck, Austria, Institute for Meteorology and Geophy-
sics and, after my retirement, lectures at the University of Hamburg and the Free
University of Berhn, where I was invited as an honorary professor.

The internal structure of the text resembles the old text of the well-known and, in its
time, excellent Handbook of Oceanography by O. Kriimmel (E. Engelhorn, Stuttgart,
Vol. I, 1907, Vol. II, 1911). This two-volume work is outdated in all of its parts and
had to be replaced in time by a completely revised modem text corresponding to our
present knowledge of the oceans. In one respect the text under consideration differs
fundamentally from Kriimmel's book since no attempt was made to deal in the
present book with historical and older work about oceanic phenomena and with
attempts to explain them in such minuteness of detail. Most of Kriimmel's material,
as will be understood, deserves at present only historical interest and would have been
for my book only unnecessary ballast. The reader who has a special historical interest
may therefore be referred to the text of Kriimmel.

Being fully aware that not aU the chapters of my work will perhaps be quite to the
taste of my oceanographic colleagues, I have always tried to present everything which
may still be of value for the further development of oceanography. I may best speed
this book to the reader with the words, splendid due to their simphcity, of the great
Newton :

"Ut omnia candide legantur, defectus in materiam tam difficile non tam reprehen-
dantur, quam novis lectorum conatibus investigentur, benigne suppleantur, enixe
rogo."

"I heartily beg that what I have here done may be read with forbearance ; and that my
labours in a subject so difficult may be examined, not so much with the view to censure,
as to remedy their defects."

"Mogen Mangel in einer so schwierigen Materie den Leser weniger zum Tadel als zu
neuen Versuchen und gefalhger Erganzung veranlassen! Um das bitte ich denselben
recht dringend."

A. Defant

Innsbruck,
March 1960

Introduction

Oceanography is the branch of science concerned with the oceans and the phenomena
occurring therein. It is a part of the sciences dealing with the Earth, and in so far as it
gives a quaUtative description of phenomena it belongs to the geographical sciences.
It uses methods essentially similar to those of the other geographical sciences and its
aim is the same as that of general geography, the classification of the different material
and energy characteristics of the phenomena found with precise definitions into differ-
ent categories and the systematic inter-relation of these. Regional geography groups
all locally co-existing and interacting phenomena on the basis of a common area of
occurrence which may be of greater or lesser extent. From the geographical point of
view there is thus a general and a regional oceanography both using principally statisti-
cal and descriptive methods.

The rapid progress of the exact sciences in recent times has led to an increasingly
rapid transition from a geographic to a geophysical treatment of the problems of
oceanography. This has given rise to a quantitative conception of oceanographic
phenomena based on physical-mathematical principles. In this respect oceanography
is a branch o^ geophysics and is recognized as an independent science comparable with
meteorology (the physics of the atmosphere) and with geophysics in its more restricted
sense (the physics of the Earth).

The history of the development of oceanography into a science is essentially the same
as that of other scientific disciplines, although it is still at a comparatively early stage.
Like all the other sciences its facts are obtained by observation. Initially these observa-
tions were made only of phenomena and conditions in the immediate neighbourhood
of continental coasts or islands. Conditions in the open sea were for long indefinite
and uncertain, and furthermore things that were new, exceptional or spectacular, were
much more interesting than normal everyday phenomena. As knowledge increased
men ceased to be content to recognise conditions and changes immediately around
them; they also sought after insight into the nature of phenomena occurring all over
the Earth. Men penetrated out into the vast stretches of the seas and there gradually
developed a conception of the ocean. The bold voyages of seamen gradually clarified
ideas of the figure of the Earth and the confirmation of its spherical shape showed the
finiteness of the oceans.

Systematic order in the collection of ships' observations and the increased accuracy
obtainable by the use of instruments came only after the beginning of the nineteenth
century. The regular navigation of the seas necessary for the expansion of trade and
commerce rapidly increased the knowledge of surface conditions which was recorded
in thousands of ships' journals of merchant marine ships. At the suggestion of the
American naval officer and oceanographer Matthew Fontaine Maury (1806-73) an
agreement was reached in 1853 at an international conference in Brussels on the form
and content of these journals, and this was supplemented by an international conference

xiii

xiv Introduction

in London in 1873. These important observations were collected in the records
of hydrographic offices or of central meteorological offices and scientifically corre-
lated. Records of temperature, salinity and currents at the surface, of tides and of the
meteorological conditions over the sea w^ere compiled, and the rapid development and
the safety of navigation can be attributed not least to this detailed knowledge of sur-
face conditions.

However, ships' journals of merchant marine ships are not sufficient to give a broad
comprehension of oceanographic phenomena. Maritime traffic is interested only in
the fastest crossing between the continents and the observations recorded in ships'
journals are confined very largely to certain routes, usually those that are as short as
possible, while the remoter parts of the oceans have been left untouched. In many
cases, however, phenomena occurring in these areas are important for a correct
scientific assessment and the comprehension of ocean phenomena in general. A know-
ledge of conditions at the surface and beneath it covering the whole oceanic space is
necessary for the further development of oceanography.

These considerations have led to the oceanographic expeditions that have contri-
buted so much to the science of the seas. The task of deep-sea expeditions is first and
foremost to determine the shape of the sea bottom and to measure as accurately as
possible the physical-chemical conditions of sea water between the bot tom and the
surface. Of major importance are the horizontal and vertical variations of the oceano-
graphic factors: temperature, salinity and dissolved gases. Variations in the first of
these indicate the variations in density and the latter ones allow a correlation with
marine biology which requires a knowledge of the environment of marine life.

In addition to this more statistical knowledge of the physical-chemical structure of
the sea it is also desirable to know something about the circulation of water masses. It
is obvious that the internal circulation of the ocean must be related to the oceanic
structure. Because the driving force for the oceanic circulation lies partly in the
movement of the air over the ocean surface and partly in regional differences between
masses of water (or diff'erences of density of the masses) due to diff"erences in tem-
perature and salinity. If conditions in the oceans are steady there will be an inverse
relationship between the circulation and the thermo-haline structure. The earliest
method used to deduce the circulation system was based on such a correlation, using
an accurate survey of the thermo-haline structure. The determination of the move-
ments of water masses, the forces causing them and their seasonal variations in time as
well as local variations and transports are the main problem of modern oceanography.

During the development of oceanography the character of oceanic expeditions has
undergone a transformation. The first expeditions were natura'ly only attempts to
clarify conditions and to overcome experimental difficulties on board the research
ships. The major deep-sea expeditions at the end of the nineteenth century and at the
beginning of the twentieth laid the foundations of modern oceanography. At first they
investigated only a section through the ocean, that is, along the route of the ship.
The results were based on discontinuous sampling and rarely reached to the sea bottom.
This method did not allow any three-dimensional conception of oceanic phenomena.

Progress in oceanographic technique on board research vessels and modern de-
velopments in the recording and interpretation of results have made it possible since
the First World War to carry out systematic investigation of the ocean, not along one

Introduction xv

or two sections but using a narrow spaced net-work of stations over the whole of a
water mass from the surface to the sea bottom. This was the logical development of the
first voyages of research ships progressing from straightforward discovery to system-
atic exploration of a whole ocean along a carefully prepared plan. After some minor
research voyages of this type by Norwegian oceanographers came the first major
expedition for the systematic survey of a whole ocean, the German Atlantic Expedition
of the "Meteor" 1925-7 (Defant, 1928).

Large expeditions such as this that give a deeper insight into the geographic variation
of the oceanographic factors over the entire ocean are essential for an extensive view
of the phenomena occurring in the oceans. The closer the network of stations the more
accurate such a survey will be, but the establishment of the closest possible network is
capable of only partial fulfilment. The work of the research ships can be intensified
only with difficulty to get a more rapid sequence of stations and there are difficulties
in the interpretation of the data recorded. The treatment of the results of a survey of
a whole ocean is based on the assumption that conditions in the ocean are steady.
However, this is only approximately true. Conditions in the water mass in an ocean
are on the whole quasi-stationary provided that they are not examined in too great
detail. Only in this case one is justified in concluding that the movements of water
masses from the thermo-haline structure, and the results of all the major ocean
surveys that have been made, have shown that this correlation of the physical-
chemical conditions can be relied upon for the general interpretation of the prevailing
currents in the ocean. This, however, gives a view of average conditions only. The
dynamics of the processes in detail are more complicated, as the somewhat rough idea
of the widely spaced network of oceanographic stations shows.

This, together with more recent theoretical considerations, has led to the conclusion
that a closer study of the dynamics of the currents in the ocean cannot be based on the
observations of a rather wide-spaced network of oceanographic stations. It will re-
quire closely knit, preferable synoptic observations which can be obtained only by
collaboration between several research ships. Apart from these more specialized
surveys of oceanographic problems the older type of oceanographic survey remains
indispensable, although modern oceanographic research will change to an increasingly
synoptic concept of oceanographic phenomena. The last international joint survey in
the Gulf Stream area north of the Azores during the early summer of 1938 (Defant
and Helland-Hansen, 1939) marked the beginning of this type of joint investigation.
Probably the largest synoptic oceanographic survey has been the Operation Cabot of
the U.S. Hydrographic Office, 6-23 June 1950, which investigated the Gulf Stream
area between Cape Hatteras and the Grand Banks of Newfoundland using six ships
(FuGLiSTER and Worthington, 1951).

This book is concerned with physical oceanography. It describes the three-dimen-
sional structure and movements, material and energy characteristics of the hydro-
sphere. Furthermore, the physical and chemical properties of sea-water, the regional
variations in the oceanographic factors and their periodic variations are dealt with.
It also describes the different types of ocean currents (ocean dynamics), and finally
the periodic movements of the water in waves, tides and related phenomena (dynamics

xvi Introduction

of periodic phenomena). The contents of this book are therefore concerned only with
general geography, chemistry, physics and the dynamics of the sea. However, outside
the scope of this book lies marine biology which is concerned with the organic life
of the oceans (plankton and fishes) which reacts not only to the external environment
but also to stimuli and incentives of non-physical origin.

Part 1

Chapter I

The Ocean

A. THE HORIZONTAL EXTENT AND THE STRUCTURE OF

THE OCEAN

1. Introduction, Vertical Structure of the Total Earth

The total Earth system can be subdivided into three parts. The solid rock forms the
Lithosphere and is the solid core on which the other two layers rest. If the rock layer
was freed from all its characteristic irregularities it would be in the geodetic sense
"flat" (a simple rotational ellipsoid). The water forming the layer next in density, the
Hydrosphere, would cover as a single ocean the entire surface of the Earth. This is not
the case. The lithosphere is very uneven, and large depressions and elevations disturb
its regular shape. There is not suflScient water to cover all these irregularities entirely,
but it fills the depressions between the continental plateaus and leaves uncovered the
upper parts as continents. This outlines the form of the lithosphere and gives the Earth
its characteristic appearance.

The third major part of the total Earth, the Atmosphere, lies as a gaseous envelope
above the hydrosphere and touches the lithosphere only over the continents. It should
be remembered that this is in fact exceptional, occurring over little more than a quarter
of the surface of the Earth. Normally, the lithosphere, the hydrosphere and the at-
mosphere are arranged one above the other with the diff'erent strata of each layer
arranged in order of density by the force of gravity. This is a necessary condition for
the static stability of the three parts of the total Earth,

The transition from one layer to another is finite and rather abrupt. The water
masses of the ocean are bounded by two main surfaces (Fig. 1, Defant, 1940).

(a) The interface between the lithosphere and the hydrosphere is the sea bottom:
across it there is a density change from approximately 2-5 to 1 -06 g/cm^. The investiga-
tion of the morphology of the sea bottom is one of the main tasks of oceanography.

{b) The interface between the ocean and the atmosphere is the sea surface; here the
density change is from about 1-03 to 0-0013 g/cm'^. All phenomena affecting both,
ocean as well as atmosphere, take their origin from this surface. An accurate knowledge
of its form is of the greatest importance to oceanography.

The water masses of the ocean lie entirely within these surfaces, forming a single con-
tinuous mass. All the energy absorbed by the ocean or given off by it must pass through
these boundaries, and this energy entering or leaving the ocean is the basic cause of all
the phenomena and changes of state in the water mass.

2. The Horizontal Extent of the Ocean and its Boundaries

The incomplete covering of the surface of the Earth by the ocean separates it into

The Ocean

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- 3.000

- ZOQO

ATMOSPHERIC

-SS'C 0.00 038 g cm"'

ATMOSPHERK

STRATOSPHERE

TROPOSPHERE

Fig. 1. Diagrammatic representation of the main boundary surfaces in tlie structure of the

Earth and the density changes at each. The figures at the right are the heights or depths in

metres above or below sea level.

land and sea. The coastal limits of the continents projecting above the surface of the
ocean are known almost everywhere with satisfactory accuracy. It is only in the polar
regions where vast areas of land are buried under ice that it is difficult to determine
accurately the limits between continent and sea. These uncertainties have recently
been considerably reduced, however. Apart from this reservation, of the 510-01
million km^ of the Earth's surface not less than 361 -1 million km^ is ocean and only
148-9 milHon km^ is land (Kossinna, 1921). The ratio of land to sea is 1 : 2-43 or 29-20
relative to 70-80%. The uncertainty in these values is not more than a few hundredths.
The Earth's surface is thus mostly oceanic. Similar relationships hold for the Northern
and Southern Hemispheres taken separately: in the Northern Hemisphere 60-7% water,
39-3% land; in the Southern Hemisphere 80-9% water, 19-1% land. Water still pre-
dominates in the Northern Hemisphere, while in the Southern Hemisphere land is
very markedly in the minority. A great circle can be drawn dividing the surface of the

The Ocean 3

Earth into a land and a water hemisphere, one containing the largest possible land area
and the other containing the largest possible water area. The pole of the land hemi-
sphere lies at 47-25° N., 2-5° W. near the mouth of the Loire, and this hemisphere
contains 52-17°o sea and 47-3% land, corresponding to a ratio of 1 : 0-90; the water
area is still shghtly greater than that of the land. The centre of the water hemisphere
lies at 47-25° S., 177-5° E., south-east of New Zealand, and this hemisphere contains
90-5° 0 water and 9-5% land corresponding to a ratio of 1 : 0-11; shghtly less than
10° o is land. For many phenomena affecting the Earth as a whole this division into
land and marine sides is of some importance.

The distribution of land and water areas given in percentage is very irregular and
apparently completely asymmetric. Table 1 gives the percentages of land and sea in
zones of 5° of latitude.

Table 1. Distribution of sea and land for zones of 5° of latitude
(In per cent, according to E. Kossin-na, 1921)

Latitude

Northern Hemisphere

Southern Hemisphere

zone

Water

Land

Water

Land

90-85°

1000

85-80°

85-2

12-8

100-0

80-75°

77-1

22-9

10-7

89-3

75-70°

65-5

34-5

38-6

61-4

70-65°

28-7

71-3

79-5

20-5

65-60^

31-2

69-8

99-7

0-3

60-55°

450

550

99-9

55-50°

40-7

59-4

98-5

1-5

50-45°

43-8

56-2

97-5

2-5

45^0°

51-2

48-8

96-4

3-6

40-35=

56-8

43-2

93-4

6-6

35-30=

57-7

42-3

84-2

15-8

30-25°

59-6

• 40-4

78-4

21-6

25-20°

65-2

34-8

75-4

24-6

20-15°

70-8

29-2

76-4

23-6

15-10°

76-5

23-5

79-6

20-4

10- 5°

75-7

24-3

76-9

23-1

5- 0°

78-6

21-4

75-9

241

90- 0°

66-66

39-34

80-92

19-08

90° N.-90° S

r total ocean

361-059 X 10'' km2, 70-80'

\ total continents 148-892 x 10^ km^, 29-20%

The thin dotted lines in Fig. 2 for 50% and 25% land show that land predominates
only in two places, between 70° and 45° N. across the Eurasian and North American
continents and at about 70° S. in the region of the Antarctic continent. In the Southern
Hemisphere, with the exception of the polar area, the land is nowhere more than 25%
of the total area. Between 55° and 65° S. the ocean forms a continuous belt around
the Earth, a fact which is of fundamental importance for many oceanographic phe-
nomena.

77?^ Ocean

Fig. 2. Percentage distribution of water and land areas in five degree zones.

The arrangement of the continents outlines the irregular distribution of the sea.
The sea fills the depressions between the continents as far as its volume allows. On
closer inspection a division into three major oceans can be recognized: the Atlantic,
the Pacific and the Indian Ocean. They are all connected with each other, forming a
continuous ocean belt in the higher latitudes of the Southern Hemisphere. This can
be seen very clearly on Steinhauer's star projection centred on the south pole. Here
the Atlantic and the Indian Oceans appear as very large and extended bays radiating
out from the circumpolar Southern Ocean (Fig. 3).

The main boundaries of the three oceans are fixed in the first place by the conti-
nents. Conventional boundaries are necessary only to the south of Australia, South
America and Africa where distinct morphological boundaries are missing. These have
been fixed by international agreement (Intern. Hydrogr. Bureau, Monaco, 1937;
WiJST, 1939).

The three major oceans are subdivided by the continental coast lines which in some
places are remarkably irregular. There is a particularly marked contrast between the
open ocean and the seas enclosed between mainland and groups of islands. The sea
areas which are separated from the ocean and project to a greater or lesser extent into
the continents are denoted adjacent seas, and according to the degree of separation
from the open ocean they may be either marginal or mediterranean seas. The demarca-
tion from the ocean is usually topographical. The more important adjacent seas are
listed in Table 4 (see p. 1 7), together with the area, the mean and maximum depths of the

The Ocean

Fig. 3. Steinhauer star projection to show the distribution of oceans and continents.

three major oceans (with and without adjacent seas) as well as for the marginal and
mediterranean seas (Kossinna, 1921; Landolt-Bornstein, 1952, article by Dietrich,
p. 460).

3. Sea -level and its Variations. Chart Datum

The surface of the ocean which forms the boundary between the ocean and the
atmosphere is in a physical sense a free boundary that may assume different forms at
different times under the influence of various internal and external forces. This bound-
ary surface is called the "sea-level". If the Earth was covered entirely by a homogeneous
ocean unaffected by atmospheric phenomena such as winds and atmospheric pressure
or the tidal forces of the sun and the moon, then there would be only a single force
acting on the sea : gravity. In the equilibrium state there can be no component of the
force of gravity along the surface of the sea and the direction of the force of gravity
must be perpendicular to the surface. This "ideal" sea-level is thus a geopotential
surface or a gravitational equipotential surface. If minor variations in the force of
gravity due to the irregular distribution of the mass of the outer crust of the Earth are
disregarded, the ideal sea-level will coincide with the surface of a rotational ellipsoid.
Even if the sea does not cover the entire Earth, the ideal sea-level will correspond to
the surface of this rotational ellipsoid. When the small irregularities in gravitational
force due to the irregular mass distribution of the Earth crust are taken into account.

6 TJie Ocean

the sea-level as a geopotential surface will no longer have the same simple ellipsoidal
form but will show little variations to either side. This irregularly shaped surface is
called in the theory of the Earth figure the "geoid". The geoid can be regarded as dis-
placed from the surface of the rotational ellipsoid by the distortions of the continental
masses. The geoid rises on passing from the sea towards the continents and falls on
passing towards the sea again. Figure 4 illustrates the undulations of the geoid around

Ocean Continent

Rototionol ellipsoid ^^llXlIIJilL-J 'n^^^^^

Fig. 4. Undulations of the geoid about the rotational ellipsoid.

the rotational ellipsoid. The ideal sea-level (geoid) lies below the rotational elHpsoid in
sea areas and above it in land areas. The magnitude of these deviations depends on the
magnitude of the gravitational anomahes in the upper crust of the Earth. It was at first
thought from theoretical considerations that the undulations of the geoid must be
rather large. However, it was found that due to the almost perfect isostatic adjustment
of the masses of the outer crust (hydrostatic equilibrium), these remain rather small
and amount to not more than rhlOOm. The forces that cause periodic variations
of the actual sea-level from the geoid were mentioned above. Amongst these are the
forces due to the attraction of the sun and the moon which produce the tides in the
ocean and the tangential force of the wind on the surface of the sea which causes
ordinary sea waves. Both of these effects on the sea-level initiate waves that can be
considered as oscillations to either side of a mean sea-level. It can be fixed at any
coastal station by continuous observation of the water level, because the influence of
the tides can be excluded if full-yearly observations are available while the effect of
the ordinary wave motions disappears in a daily mean of observations.

Other forces affecting the ideal sea-level may cause long lasting displacements of the
actual sea-level from the geoid. If these forces are steady the corresponding displace-
ments will also be steady and give a static equilibrium state. Also in the case of slowly
changing forces the time will be sufficient for the sea-level to follow the changes. If,
however, there are rapid changes in the intensity of the force the situation will be
more complicated and an oscillation may develop depending on the size of the water
masses involved.

An important source of steady displacements of this kind from the ideal sea-level
is the effect of barometric pressure. The ocean reacts to steady changes in the atmos-
pheric pressure on the surface like an enormous water barometer: as the atmospheric
pressure rises the sea-level will fall below the geoid, as the atmospheric pressure falls
it will rise above it. When conditions are stationary there can be no pressure difference
between two points at the same level within the ocean. The pressure at a depth h^
below ideal sea-level in a homogeneous sea of density pq will be

The Ocean 7

where p^ is the air pressure at the surface, g is the gravitational acceleration and |^o
is the deviation of the surface from ideal sea-level. At another place it will be

P = Pi + gPi(fh + O-
The pressure difference between the two places will then be

^P = -g(po - PiVh - gipo^o - Pi^i)- (I-O

For a completely homogeneous sea (pq = pi) the relative deviation of the sea-level
from the geoid will be

J^=-A.zlp. (1.2)

If the average density for sea-water is taken as 1 -028 then

J I in dynamical cm = — 0-973/1/7; (Ap in mbar),^

}■ (1.3)

J ^ in cm = —0-993Ap; (Ap in mbar). J

The numerical factor in the last equation will be 1 -326 when Ap is expressed in mm Hg,
because 1 millibar (mbar) corresponds to 0-75 mmHg.

For a steady difference in air pressure the displacement of the sea-level from the
geoid in cm will be 0-993 times the local variation in atmospheric pressure measured
in mbar, in the opposite direction. From a knowledge of the steady pressure distribu-
tion at sea-level the deviation from ideal sea-level can easily be found. In January the
barometric pressure in the high-pressure cell near the Azores is about 1020 mbar, in
the Icelandic low-pressure area it is about 990 mbar. It can therefore be expected that
the sea-level in the area of the Irming Sea will be about 30 cm higher than at the
Azores. Comparisons between changes in barometric pressure and changes in sea-
level made at polar stations, where the ice covering allows them to be followed more
easily, have shown satisfactory agreement between observed and calculated values of
sea-level (Hessen, 1931; Wegener, 1924).

Other effects due to the inhomogeneity of the water in the ocean and to the currents
associated with it, and also to phenomena caused by the blocking of ocean currents
at continental coasts (water level rise, Anstau) are harder to deal with. All these aperi-
odic stationary deviations of the actual sea-level from the ideal are included in the
concept of the physical sea-level. This physical sea-level is, under steady conditions, the
true boundary between the ocean and the atmosphere.

The methods used to fix the position of the physical sea-level relative to the surface
of the geoid will be described later (Part II). The effect by itself of different distributions
of density in different water masses within the ocean can be found using equation (I.l).
Assuming the barometric pressure being the same at both stations (Ap = 0) it follows
approximately

where h^ is the depth at which the pressure difference within the water mass vanishes.
For a density difference of 10~^ and a water volume of 100 m vertical extent, the

8 The Ocean

lighter of the water masses must be 10 cm higher than the heavier. If the density differ-
ence changes with the depth the above equation will include the integral of

Po — Pi

taken from the surface to the depth /;.

For practical purposes the mean water level is determined at coastal stations by a
tide gauge. Calculation of a mean value will eliminate the periodic factors (tides and
waves) but other factors will remain ; in the first place the aperiodic changes in mete-
orological factors such as the wind, barometric pressure, precipitation and evaporation
that can only be eliminated by taking a mean value over a number of years. However,
even this mean value cannot be taken as invariable. It will reflect secular (long period)
changes in meteorological factors and also slow deformations of the Earth and slow
changes in the total water mass of the oceans. For comparison and inter-relation of
mean sea-levels fixed at different places along a coast, precision levelling between these
points is essential. This must be taken over land and be independent of the conditions
in the sea in order to show whether the mean sea-levels are in one and the same or in
different niveaus. On the subject of precision levelling along the Baltic coast (1896-8)
see Westphal (1900), along the east coast of North America see Anvers (1927) and
Bowie (1936), and on the interpretation of these see Dietrich (1937).

Sea-level at almost all coastal stations shows clearly an annual period which is
related principally to wind conditions along the adjacent sea coast; thus the sea-level
at Aden is connected with the monsoon in the Arabian Sea (Krummel, 1907), while in
Japanese waters the annual changes in barometric pressure and in density of the water
are of greater influence (Nomitsu and Okamoto, 1927). On the annual variation in the
sea-level along the Baltic coast see Hahn and Rietschel (1938), and Bergsten (1917).
Along the coasts of those seas where there are strong tides the determination of mean
sea-level is more complicated since the effect of the tides has first to be eliminated.
This is best done by subtracting the mean tide level calculated by means of the har-
monic tide constant from the actual change in water level as shown by the tide gauge.
The remaining part is the aperiodic deviation in water level (in addition possibly free-
oscillations of water masses) which must be related to meteorological factors (Marmer,
1927). If this ideal method is not possible the mid-point of each tide can be found by
taking an average of hourly readings over a full tide period and it can be assumed that
this value is reasonably free from any cosmic influence. An investigation of this type
has been carried out for the German Bay (North Sea) by Leverkinck (1915).

The changes in sea-level recorded on a tide gauge can also be simulated by a rise or
a fall of the land on which the gauge stands. Movements of the coast line forming the
boundary between land and sea may be compounded of two movements, those of the
water and those of the land (Penck, 1934). As the ocean may be compared with a
large vessel filled with water, changes in the water surface may arise through changes
in the volume of water in the ocean or by alteration of either its size or the position
of the water surface in the vessel. All changes in sea-level that affect the entire ocean
surface in the same direction are termed, following Suess (1888), eustatic. This in-
cludes two very slow changes: the nomic and \h& juvenile motion. The first is due to the
slow erosion of the land that lifts the sea bottom, the second is due to the continuous

The Ocean 9

addition of juvenile water from the interior of the Earth by volcanic and thermal
activity.

According to Penck, about 12 km^ of solid material are carried into the sea annually
and this would raise the level of the sea by about 33 mm in a thousand years. This
nomic movement will continue as long as there is land that can be eroded. When this
final state of erosion has been reached the sea-level will have risen about 250 m
higher than it is at the present time. The juvenile increase in the level of the sea amounts,
according to Penck, to not more than about 2-8 mm in 1000 years or barely one-
twelfth of the nomic. It will continue as long as volcanic activity on the Earth persists.

A faster change than either of these eustatic movements is that due to the melting
of glaciers. During the ice ages there was approximately 40 miUion km^ more ice
covering the land than there is at the present time. This melted during a period of
10,000 to 20,000 years and raised the surface of the sea by 100 m or by 5-10 m in
1000 years (Ramsay, 1939; Penck, 1933). Melting of the present-day ice of glaciers
covering the land (22-2 million km^) would raise the sea-level by 55 m. The level of
the ocean varied during the ice ages over a maximum range of 155 m.

The movements of the solid crust of the Earth may be of either tectonic or volcanic
origin or they may be due to isostatic elevation or subsidence of single parts of the
crust. The first may be accompanied by considerable local changes in a short time.

Chart datum. Sea charts showing depths at different places give a picture of the
topography of the sea bottom. These depths are not calculated from sea-level (as a
reference level) but from a so-called chart datum. This has been done for purely
practical reasons concerned with navigation. Chart datum on English and German
charts is that of mean low-water springs; on French charts it is the level of the local

I Nash Point I
0

Portishead

0 5 10 15

Fig. 5. Mean sea level and chart datum in the main shipping route in the Bristol Channel.

Dungeness g^-^ ^^^

\ Le Colbart

\ J,

0 5 10 15 Sm

Fig. 6. Mean sea level and chart datum in the straits of Dover.

The Ocean

lowest low water and on American charts it is the level of local mean low water.
Only the sea charts of tideless mediterranean seas relate their depths to mean sea-level
(e.g. the Baltic). Chart datum is nowhere the same as normal datum {NN) for carto-
graphical surveys on land but is generally lower. Since the tidal range varies from one
coastal station to another the chart datum forms an undulating surface which in
general falls as it approaches a coast. This fall is greatest in funnel-shaped bays where
the tidal range rapidly increases towards the inner end. On the open sea there are only
small differences between chart datum and mean sea-level.

Chart datum must be taken into consideration in more accurate hydrographic cal-
culations. Raverstein (1886) first pointed out the importance of chart datum and
prepared two charts of a part of the English Channel. One of these showed isobaths
according to the sea chart (calculated from chart datum), and the other showed iso-
baths calculated from the surface of the geoid. These charts demonstrate clearly the
importance of considering a reference level. Figures 5 and 6 show two profiles of the
differences between mean sea-level and chart datum for a longitudinal section along
the Bristol Channel and for a cross-section of the Straits of Dover. For further informa-
tion on the often very complex question of chart datum see especially Horn (1944).

B. THE THREE-DIMENSIONAL STRUCTURE OF THE OCEAN

1. Methods of Recording Deep-sea Data

The safety of shipping in coastal waters requires an accurate topographical survey
to considerably greater depths than the 12 m draught of the biggest ships, usually
down to 200 m. This is about the maximum depth at which soundings can be made
with any accuracy using a hand lead line. Soundings taken in this way can also be used
to measure the depth of water under a vessel anchored in shallow water and hence to

'feeeste" StationM

«.

* *.

•x •

* J

• " ^ ,

• :,

* K •

• 1

■

»• 1

•

• K

I 1

1 1

\ '-

: 1 1

■ ««

13-

8 10 12 14 16 18 20 22 24 2 4 6 8 10 12
I7-2II-34 hr

Fig. 7. Tides determined by sounding from an anchored ship
(0 = 53° 55-9' N, A = 7= 52-2' E).

determine the range of the tide at that point. This is the simplest method of determin-
ing the tidal range at a distance from the coast in adjacent seas that are not too deep
and along the continental shelves. The hemp lead line should have a piano wire trace
at the upper and lower ends. Soundings of this type can, with some practice, be de-
termined to within ±5 cm even for wave motion. Figure 7 gives an exam.ple of a tidal
cycle measured in this way at a station in the southern North Sea.

The Ocean 11

Soundings at depths greater than 200 m cannot easily be made with a hemp line
by hand since the weight of the line and lead is too great and it is difficult to feel the
contact with the bottom. The measurement of greater depths is extremely difficult and
it took several decades of experimental work before deep-sea soundings could be made
reliably at any point in the ocean.

The measurement of the depth of the sea (Stahlberg, 1920) is the determination of
the perpendicular distance between the surface and the sea bottom. At great depths
this is difficult because: (1) the bottom contact is not easy to detect, and (2) hauling
in the increased sounding weight is very laborious unless it is done by machine. Two
conditions are necessary for a reliable deep-sea sounding: (1) the use of a thin steel
wire in place of the hemp line used previously, and (2) the release of the sounding
weight on contact with the sea bottom.

The wire sounding method used at great depths will not be described in detail here
since it is essentially a technical question. Further details can be found in technical
handbooks [see especially Pratje (1952), and Oceanographic Instrumentation (Re-
port of conference, Rancho Santa Fe, Cahfornia, 21-23 June 1952)].

The development of echo sounding has revolutionized the investigation of sea-
bottom topography; wire soundings could never have been made in such large
numbers nor have given such good results for the rapid and precise elucidation of
conditions at the bottom of the ocean, and centuries would have been needed to get
the results that can be obtained without difficulty in a few years by echo sounding.
The basic principle of echo sounding is very simple; it measures the time required for
a sound wave to travel from the bottom of a vessel (the sea surface) to the sea bed and
back. The returning wave can be detected as an echo and amplified. To calculate the
depth, knowing the speed of sound in sea water, it is only necessary to determine the
time from emission of the sound until the echo is detected- — the echo time. If the time
is /, the speed of sound in water v and the depth of the sea h, then

Echo sounding makes it possible to sense the bottom of the sea accurately and to
ascertain its actual topography. A vessel equipped with echo sounding can fix the
depth of the sea without loss of time while moving at full speed. Scientifically, sonic
sounding is of value only when: (1) it is combined with an accurate determination of
the position of the vessel which in general should not be determined less accurately
than ± 1 nautical mile and (2) when the mean velocity of the sound emitted by the
echo sounding apparatus is known in addition to the echo distance. Only then is it
possible to convert the value obtained to the true depth. The enormously increasing
number of echo soundings requires the establishment of an international office to
correct and unify the mass of data and to chart it after critical interpretation. This
would give results of great utility both scientifically and for the improvement of the
sea charts of all nations (Defant, 1938).

Echo sounding has only one disadvantage compared with wire sounding; it cannot
be combined with the collection of bottom samples which are necessary to ascertain
the nature of the bottom sediments. If these are needed wire sounding is indispensable.
However, it is possible with more modern types of echo sounding equipment to draw
some conclusions about the nature and thickness of the bottom sediments from the

12 The Ocean

appearance of the echo in the receiver. The structure and form of the returning wave is
dependent on the nature of the reflecting surface. If the oscillatory form of the re-
flected wave can be ascertained in the receiver it is possible to decide whether the
bottom is rock, sand, mud, or other material. It is very frequently found that the echo
is split into broader or narrower bands which are clearly connected with the different
layers in the bottom sediment (mud or rock). The echo sounder thus gives a pre-
liminary idea of the nature of the bottom and often the thickness of the soft upper
sediment. This was first mentioned by Stocks (1935). For further details reference
may be made to Evving, Crary and Rutherford (1917), Bullard (1938) and
EwiNG and Vine (1938). Another method of studying the structure and thickness of
the deep sea sediments has recently been developed by Weibull (1947). Very good
results were obtained with this by the Swedish "Albatross" Expedition (Pettersson
1946).

Indirect depth determination with an unprotected reversing thermometer. Ruppin
(1906. 1912) first suggested the use of the difference between protected and unpro-
tected reversing thermometers for the measurement of the depth at which the reversing
frame or the water sampler on which the thermometers are mounted is reversed. The
usefulness of the method has been shown by the investigations which he carried out at
depths up to 100 m and by those of von Perlewitz at up to 1000 m. Brennecke (1921)
on the "Deutschland" Expedition of 191 1-12 made valuable use of it, and it was used
systematically for the first time on the "Meteor" Expedition of 1925-7 (WiJST, 1932).
In both wire sounding and in oceanographic serial observations there is always a wire
angle of greater or lesser magnitude and it is therefore extremely valuable to have a
method available which allows a reduction of the temperature and salinity values to
true depth or which ascertains a determination of depth independent of the wire angle.

For the construction and function of the reversing thermometer, the corrections
applied and the accuracy of the depths obtained [see particularly Oceanographic
Instrumentation (Report of conference, Rancho Santa Fe, Cahfornia, 21-23 June
1952, p. 55)].

2. The General Morphology of the Sea Bottom

The topography of the bottom of an ocean or part of an ocean can be conveniently
shown on a depth chart on which all available soundings are recorded after critical
interpretation. The reliefs of the sea bottom can be shown by drawing lines of equal
depth (isobaths) at fixed intervals. Constructing the isobaths between separate soundings
isessentially a question of interpolation which is considerably facilitated if the soundings
are distributed as evenly as possible over the whole area. This condition is unfortu-
nately very rarely satisfied, even less so after the introduction of sonic sounding.
Apart from the more sporadic distribution of earlier wire soundings there is now a
greater concentration of soundings along isolated fines of echo soundings resulting
in an extremely uneven distribution of depths and, while some parts are extremely well
surveyed, there are very large areas with only single soundings. The task of preparing
isobaths for an entire ocean has thus become more difl[icult than before the introduction
of echo sounding.

The construction of the isobaths for an ocean area depends on subjective considera-
tions; the lines must of course be fitted to the soundings, but the available points

The Ocean 13

usually allow considerable elbow-room for the use of ideas and speculations on the
bottom topography afforded by other knowledge (for example, geological). In par-
ticular, the construction of the isobaths requires good use of oceanographic view-
points. The distribution of temperature and salinity at the sea bottom and in the
water immediately above it are dependent on the bottom topography and often
allow greater accuracy than is possible from the records of depths alone, for example
in the determination of depths on saddle points or the position of cross-ridges and
others. Indicators such as these of the course of the isobaths are always valuable and
deserve full attention. In this connection, see especially Stocks and Wust (1935) in
the addenda to the chart of the Atlantic Ocean in the "Meteor" volumes.

Good charts are not available at the present time for all the oceans and adjacent
seas; it is to be expected that there will be considerable improvement here in the future.
Apart from the older depth charts in the Sailing Directions for single oceans and
charts produced by single expeditions the following may be noted :

(1) The Carte Generale Bathymetriqiie des Oceans, scale 1 : 10 million, produced
by the Hydrographic Bureau in Monaco; 16 sheets on Mercator projection:
second edition, 1911-30, third edition from 1935.

(2) The ocean chart published by Groll (1912) in which all depths available up
to that time were interpreted in a uniform way and used for careful construction
of the isobaths; equal-area projection on a scale of 1 : 40 million,

(3) The chart of the total Atlantic Ocean on the records of the "Meteor"; a general
chart, 1 : 20 milHon on the Lambert equal-area azimuthal projection with iso-
baths at 500 m intervals (Stocks and Wust, 1935). In addition to this there is a
basic chart of oceanic soundings on a scale of 1 : 5 million in 1 3 sheets (4 sheets
published, Stocks, 1937) showing all the critically checked soundings in this
ocean.

(4) A more recent chart of the Indian and Pacific Oceans has been given by Schott
(1935) on an equal-area projection, on a scale of 1 : 60 million, with the nature
of the bottom topography of these oceans indicated with sufficient accuracy.

(5) An excellent chart of the sea bottom topography of East-Indian Seas was con-
structed by VAN RiEL (1934) and was published in the scientific results of the
"W. Snellius" Expedition.

For more recent charts of parts of the oceans and adjacent seas, see the sections on
the special morphology of these areas. The charts accompanying this book(Plate 1) give
a summary of what is known of the main features of bottom topography of the oceans.
Much of the knowledge obtained by more recent expeditions by echo sounding has
been taken into consideration here, in so far as the small scale will allow. In these
charts the isobaths are drawn for every 1000 m and the 200 m isobath has been shown
where the scale permits to show the limits of the continental shelf. The coloration
of the depth-intervals gives a clear picture of the general bottom topography in spite
of the confusion of fines at some points. In order to make the characteristic bottom
configurations such as deep-sea basins, troughs and ridges and of the cross-ridges,
deep-sea canyons and other forms which may occur, more visible, a somewhat
schematic chart has been prepared and is reproduced in Plate 2 (Defant, 1947). All
the important peculiarities of bottom topography of the ocean have been indicated by
letters and numbers.

The Ocean

The first scientific interpretation of the topographical chart of the ocean bottom
taken in conjunction with a contour map of the land areas of the Earth was a general
investigation of the relationships of heights and depths on the surface of the Earth crust.
This was a purely statistical analysis of the variations of the surface of the solid crust
about an average value, the mean crust level. If the whole of the solid crust of the Earth
were levelled off" to give a single solid sphere, the mean level of the solid surface would
be 2440 m below the present sea-level. The level of the sea itself would then be
about 260 m above the present level, that is, the solid crust would be covered by a
layer of water 2700 m thick (Kossinna, 1921). It would be expected that the fre-
quence of occurrence of individual heights and depths was entirely random. The mean
crust level (taking the present sea-level as zero: —2440 m) should occur most fre-
quently, and the frequencies of individual heights and depths around this should
form a probability curve. In these chance cavities the water would collect as oceans
and the formation of the oceans would then offer no problems, since they would
obviously form in the deepest depressions of the crust.

The statistical distribution of the heights and depths of the Earth crust has, however,
led to the striking result that the frequency in no way approaches a Gaussian-probabi-
lity curve. On the contrary, there are two height-intervals which occur with high fre-
quency while the other, less frequent, intervals group themselves around these two
culmination points as two probability curves (Fig. 8, Table 2).

6000

4000

2000

E 4000

6000

Sea level

- ^

Aver(

]ge crust

level

r""^

! 12 16

Frequency percentoge

Fig. 8. Frequency distribution of different height and depth intervals over the entire surface

of the Earth.

The two maximal frequencies lie at the height-interval of 0-1000 m and at a depth
interval of 4000-5000 m; nearly 45% of the entire surface of the Earth falls within
these two intervals, while only 10% falls on the other eleven steps. It is especially
noticeable that the mean crust level of —2440 m (depth interval —2000 m to —3000 m)
occurs infrequently, and is indeed very near the minimum between the two maxima.

The Ocean

Table 2. Frequency and areas of individual height- and depth-intervals of the earth crust

(According to Kossinna, 1921)

Interval
(m)

Areas
(10« km-)

Per cent

Interval
(m)

Areas
(10' km^)

Per cent

>5000

0.5

>5000

0-5

4000-5000

2-5

0-5

>4000

0-6

3000-4000

0-6

>3000

1-2

2000-3000

>2000

3-2

1000-2000

4-7

>1000

7-9

1000-500

27-]

5-3^

6-5 yii-i

> 500

13-2

500-200

33 ;^108

48j

> 200

100

19-7

200-0

9-4J

> 0

148

28-1

0 200

fs.iy^

t^y^

>-200

176-5

33-7

- 200- -1000

>-1000

192

36-7

-1000- -2000

2-9

>-2000

207

39-6

-2000- -3000

24-5

4-8

>-3000

231-5

44-4

-3000- -4000

13-9

>^1000

301-5

58-3

-4000- -5000

119

23-3

>-5000

421-5

81-6

-5000- -6000

16-5

>-6000

505-5

98-1

>-6000

4-5

0-9

> -10,000

5100

1000

The position of the two maxima can be fixed more closely by investigation of denser
intervals. It is apparent that one maximum falls within the interval 0-200 m and the
other within the depth interval 4600-4800 m. The structure of the crust thus in-
cludes two special areas: (1) a land area with a height of 100 m to 200 m, and (2) a
sea area at a depth of about 4700 m. These two areas together include almost 65%
of the entire surface of the Earth. These relationships can also be shown in another
way in the "Hypsographic curve for the surface of the Earth" (Fig. 9) which depends
on the areas in each separate height- and depth-interval over the surface of the Earth.

10
8
6

4
I 2

i 0

Q
-2

-4

-6

-8
-10

Average level of the physicol earth surface J +_245m

Continental block

I I I I I I I I I

^Continental slope I

■ 1270m I I

Average crust level".- 2.440 m

I M I I I I I I

Average ocean depth:— 3800 m

100

400

200 300

mill, qkm

Fig. 9. Hypsographic cur\e for the surface of the Earth.

The Ocean

This shows a stepwise form and is divided by four inflection points into five parts which
may be regarded as natural regions of the land and of the sea:

(1) Summits. All land above 1000 m (approx, 40 million km-, mean height 2040 m,
maximum height: Mount Everest 8882 m)

(2) Continental plateaus. Land below 1000 m and the continental shelf to —200 m
(approx. 136 million km^, mean height 230 m)

(3) Continental slope. From the edge of the shelf at —200 m to mean crust level
—2440 m (approx. 39 million km^ mean depth 1270 m)

(4) Deep-sea bottom. Sea bottom from —2440 to —5750 m (approx, 284 million
km', mean depth 4420 m)

(5) Deep-sea depressions and trenches. Below —5750m (approx. II milhon km^
mean depth 6100 m, greatest depth: "Emden" deep in the Philippines trench
10,800 m).

This marked distribution into high and low areas divides the surface of the Earth
into: (1) a high continental block which includes all land areas, the adjacent and parts
of the marginal seas and the continental shelf and projects about 3100 m above the
mean crust level, and (2) the deep sea which lies in basins in the Earth's crust whose
bottom is about 2000 m below the mean crust level. The division of the Earth's
crust between the continental block and the deep sea is shown in the summary in
Table 3 and is illustrated schematically in Fig. 10. These show clearly the sharp division
between the two parts: the continental block and the deep sea; the continental slope

Table 3

Oceans

per
cent

per cent of total
Earth surface

3611xl0»km2

70-8% of the
Earth surface

Adjacent and

' Shelf

43-7

3-51

mediterranean seas:

400xl0«km2

Continental

31-8

2-5 1 7.9

7-9% of Earth ^

slope

surface |

Deep sea

24-5

1-9J

' Shelf

2-7

1-7']

Oceans:

32Mx]0'km2

62-9% of 1

Continental
slope

4-8

>62-9

Earth surface

Deep sea

92-5

58-2]

JO-8

Continents

148-9 X 10" km2
29-2% of the
Earth surface

29-2

Total

1000

Total deep sea: 601%; Continental plateau (continents plus shelf): 34-4%;
Continental slope: 5-5 %

1000

The Ocean

Fig. 10. Schematic representation of the Earth's crust by a continental block and a deep sea.

Table 4. Area, volume and mean depth of oceans and seas
(For Atlantic Ocean according to Stocks 1938, otherwise according to Kossinna 1921)

Body

Area

Volume

Mean depth

Greatest depth

(10« km2)

(10« km^)

(m)

Atlantic Ocean

106198

353-498

3331

85261

Indian Ocean

74-917

291-945

3897

7450!

Pacific Ocean

179-679

723-699

4028

1 0,800 §

Atlantic Ocean (excluding adjacent seas)

82-216

318-078

3868

8526

Arctic Mediterranean

14-057

21-453

1526

5180?

American Mediterranean

4-311

9-373

2174

6269

Mediterranean Sea and Black Sea

2-969

4-318

1458

4404

Baltic Sea

0-422

0-023

463

Hudson Bay

1-232

0-158

128

229

North Sea

0-575

0-054

665

English Channel and Irish Sea

0178

0-010

263

Gulf of St Lawrence

0-238

0-030

127

549

Indian Ocean (excluding adjacent seas)

73-443

291030

3963

7450

Red Sea

0-438

0-215

491

2359

Persian Gulf

0-239

0-006

Andaman Sea

0-798

0-694

870

4177

Pacific Ocean (excluding adjacent seas)

165-246

707-555

4282

1 0,800 ii

Asiatic Mediterranean

8-143

9-873

1212

6504

Bering Sea

2-268

3-259

1437

4273

Okhotsk Sea

1-528

1-279

838

3374

Japan Sea

1008

1-361

1350

3712

East China Sea

1-249

0-235

188

2377

Gulf of California

0162

0-123

813

2274

Bass Strait

0-075

0-005

—

All oceans (including adjacent seas)

361-059

1370-323

3795

—

t Puerto Rico trough north of Puerto Rico.
% Java trench, south of Java.

§ Philippines trench north-east of Mindanao ("Emden" depth).

li Mariana trench, about 11 ° N., 143° E. Gr. greatest depth 10,363 m (according to Cabruthers
and Sawfori, 1952).

77?^ Ocean

includes not more than 6% of the surface of the Earth, and this percentage is being
decreased rather than increased by the results of echo sounding. These figures empha-
size that the deep-sea basins are not just chance depressions in the crust of the Earth.
This division of the structure of the Earth is one of the most important of geophysical
phenomena and requires a special explanation that must be very closely connected
with the history of the Earth.

Charting the sea bottom by means of isobaths and measurement of the areas of the
different depth-intervals makes it possible to calculate the volume of each ocean and
of the total ocean. The quotient of the volume and the surface area gives the mean
depth. The volume of the ocean (including all the adjacent seas) amounts to 1370-6
million km^ and the mean depth is therefore around 3800 ± 100 m. The volume
and mean depth can also be worked out for parts of the ocean and for the adjacent
seas: the values for most areas according to Kossinna are given in Table 4. The
Atlantic, Indian and Pacific Oceans have the mean depths 3930, 3960 and 4280 m
respectively. These figures are not very different; the mean deviation is little more than
4^0. In addition to this general agreement, the figures for the depth-intervals in all
three oceans, as shown in Table 5, demonstrate a very similar morphological structure
of the Earth crust. This is further proof of a uniform structure in different parts and
an indication that the existence of the two favoured levels of the Earth's crust repre-
sented by the continents and the deep-sea bottom is a universal phenomenon prevailing
over all parts of the Earth's crust. If the average density of sea water, taking the com-
pressibility into account, is as 1-037, the total mass of the ocean will be 1-42 x 10^^ =
1-42 trillion tons which is only 1/4200 part of the mass of the Earth.

Table 5. Morphological structure of the three oceans (exchiding mediterranean seas).

Areas of the different depth-intervals given in percentage of the total Earth surface

(Atlantic Ocean according to Stocks 1938; otherwise according to Kossinna 1921)

Depth-interval
in km

0-0-2

Atlantic Ocean
Indian Ocean
Pacific Ocean

All oceans

5-8
3-2
1-7

3-1

0-2-1

1-2

2-3

3-4

4-5

3-8

3-7

7-5

21-3

33-9

2-7

3-1

7-4

24-4

38-9

2-2

3-4

5-0

19-1

37-7

2-8

3-4

6-2

20-1

36-6

5-6

23-3
19-9
28-8

26-2

6-7

0-7
0-4
1-8

1-2

0-3
0-3
0-1

Sum

100-0
100-0
100-0

100-0

3. Special Characteristics of Sea-bottom Topography

The larger and smaller oceans and parts of the oceans are usually considered as more
or less extended volumes sunk into the solid crust of the Earth. From this one is guided
to assume that the sea bottom taken as a whole is concave inward. In reality this is so
only in exceptional cases; in general the sea bottom arches upward and follows the
surface of a sphere with a somewhat larger radius than that of the surface of the Earth.

Expressed in another way the radius of curvature of the sea bottom points towards
the centre of the Earth almost all the time and differs little from the radius of curva-
ture of the Earth. If large areas are considered, really concave basins occur very in-
frequently and are limited to the margins of the deep ocean trenches, to crater-
shaped basins and especially to individual adjacent seas. Bathymetric charts of the

The Ocean

ocean bottom, based on a few wire soundings, gave rise earlier to an impression of a
certain smoothness and evenness of the sea bottom. Especially the bottom slope
between two sounding points was ascertained and in most cases was found to be
less than the smallest deviation from horizontal that the human eye can still detect,
(a slope of 1 : 200 or a slope angle of 0° 17'). Actually, values found in this way showed
very few vertical divisions over wide stretches of the ocean. This very smooth sea-
bottom topography has, however, been shown by the much closer values given by
echo sounding to be at least partly a misapprehension caused by the small number of
wire soundings. Without doubt the sea bottom on the whole and especially away from
areas where orogenetic and volcanic forces are active is on a small scale far more
smooth and even than the surface of the land. The effects of the atmosphere, weather-
ing and erosion by running water which all contribute to the variety of small forms
which occur on land surfaces are of course all absent. However, echo-sounding pro-
files at close intervals very often show considerable bottom irregularity. All echo-
sounding profiles so far obtained are similar in this respect. The morphological inter-
pretation must be made with the greatest caution since for greater clarity the results
are usually shown with a strongly exaggerated vertical scale. Some vertical distortion
is, however, essential when the profile extends over such great distances in order to
show the details of the sea bottom clearly. Figures 1 1 and 12 show the "Meteor" profile

Echolot of "meteor" on profile 5K

5000

Fig. 1 1 . Echo sounding profile across the South Atlantic obtained by the "Meteor" at 23 "^ S.
(profile VII: 21-25-24° S.); with 180-fold enlargement of the vertical scale and disregarding

the curvature of the Earth.

Fig. 12. The same echo sounding profile as in Fig. 11 taking the curvature of the Earth into
account. Upper curve: vertical enlargement 1:3; lower curve: 1:30 (according to Stocks).

VII (21-25-24° S.) in two different forms (according to Stocks, 1936). The upper
diagram shows the echo-sounding profile along a line from Rio de Janeiro to Whalefish
Bay with a 180-fold magnification of the vertical scale and without taking the curva-
ture of the Earth into consideration. In the lower profile, on the same horizontal
scale, the curvature of the Earth at latitude 23° has been taken into account; the outer
arc is the surface of the sea, and below this the upper curve shows the sea bottom with
a vertical exaggeration of 3 : 1 while the lower curve shows the sea bottom with a
vertical exaggeration of 1 : 30. The details of bottom topography and changes of
slope are still easily recognizable on the curve with a 30-fold vertical exaggeration and

20 The Ocean

are closer to reality than that in the upper diagram. A quantitative reading of differ-
ences in height is, however, hardly possible here, and with a vertical magnification of
only X 3 the thickness of the thinnest lines on the diagram is significant. The appear-
ance of prominent features such as the Whalefish ridge can scarcely be seen and any
qualitative differentiation into areas of greater or lesser irregularity is hardly possible.
Magnification of the vertical scale is thus necessary from the topographical point of
view, but must be used with appropriate caution.

No accurate numerical evaluation of the echo-sounding profile, in order to fix the
degree of bottom irregularity in different parts of the ocean, has yet been made. The
superficial appearance of most of these profiles shows that the bottom relief varies
from one area to another, and care is needed in making generalizations as these sound-
ings give more and more detail. In most cases there is a relatively smooth bottom pro-
file in the broad extended deep-sea basins and considerably greater irregularity over
the central ridges and over the rises that separate the broad basins; considerable
elevations above the mean surface of an area occur frequently in the vicinity of great
depths and depressions so that extreme variations in depth are very often situated
close together.

Only certain especially characteristic forms of the commonly occurring typical bot-
tom features will be discussed here. Stretching out to sea from the edge of the land
there is first the beach which at high water is part of the sea bottom and, at low water,
is part of the land. This amphibious part of the Earth's surface according to the
estimate of Schott has an area of 1-6 million km^ or about 0-4% of the ocean area.
Outside this the ledge-like rim appears, sometimes narrow, sometimes broad, but rarely
completely absent, and is called the continental shelf. From the boundary between the
land and the sea the sea bottom, except along coastal cliffs, slopes gently down at a
slight angle, at the most 1-1-5°, This angle gradually increases and near the 200 m
isobath it changes abruptly to the steeper gradient of the continental slope. The mean
slope angle is about 3° here but in isolated cases it may be appreciably larger (6-10°
or more). The edge of the shelf is normally at a depth of between 100 m and 200 m,
but in some cases it appears only at a depth of 400-500 m. The continental shelf is
seldom a uniform surface. It is very frequently broken by canyons, furrows and
troughs, and shows clearly the effects of the more intense movements of the water
because of the shallow depth (ocean and tidal currents). These effects of the action of
the ocean are not found everywhere; in some places the sea bottom has clearly been
formed during the ice ages by glacial action and has the character of a drumlin
landscape as in the Irish Sea, for instance, between Ireland and Scotland.

The continental shelf can usually be regarded as a part of the continental block
which has been flooded by the sea, and its formation and topography are partly the
product of the separation of the continents through accumulation and partly due to
the erosion of the coast by wave action (Penck, 1934). Up to the present time no
detailed investigation of the extent of the continental shelf has been made. Usually
the 200 m isobath is taken as the outer limit of the shelf and the area of the shelf
within this is usually designated as "bathymetric". In his statistics of the ocean depth
Kossinna has listed these areas for each continent (Table 6). The bathymetric shelf
extends over an area of 27-5 milhon km^ or 7-6% of the area of the ocean; Wagner
has given the value 30-6 and Kegel has given 29-5 million km^. The mean depth of the

The Ocean

shelf has been estimated by Kossinna as less than 100 m and is probably between 50 m
and 70 m.

Table 6. The shelf-areas (0-200 m) of the continents and oceans respectively
(10" km^, according to Kossinna, 1921)

Continents

Areas

(excluding mediterranean seas)

Mediterranean seas

Europa

311

Atlantic Ocean 4-59

of the Atlantic Ocean 9-52

Asia

9-38

Indian Ocean 2-37

of the Indian Ocean 0-80

Africa

1-28

Pacific Ocean 2-89

of the Pacific Ocean 7-32

Australia

2-70

North America

6-74

Sum 9-85

Sum 17-64

South America

2-42

Antarctic

0-36

Sum 25-99 + rather distant islands 1-50

Sum 27-49 x IC km% 7-6 % of the sea surface

The shelf near the continental slope, often at a considerable distance from the coast,
shows remarkable canyon-like troughs stretching over the bottom of the shelf and the
adjacent continental slope. While previously only a few of these remarkable structures
were known it has been shown recently, especially by the work of the United States
Coast and Geodetic Survey, that they are of wide occurrence. Their topography can
be rapidly and accurately determined by echo sounding. They were first thought to be
drowned, sunken valleys, but it has been shown that they probably have a different
origin. Two trough forms are found : submarine valleys in areas which have at some
time been strongly glaciates (for instance around Iceland) and submarine canyons in
regions which have remained unglaciated. The latter are usually found only at the
edge of the shelf in the area of transition to the continental slope; these reach large
depths (2000-3000 m) and often have little apparent connection with the topography
of the neighbouring coastal area. Several series of these submarine canyons have been
found and accurately charted: on the continental shelf and the edge of the shelf along
the North American coast north of Cape Hatteras among which is the long-known
submarine valley of the Hudson mouth (Fig. 1 3), along the coast of Cahfornia and along
the coast of Washington and Vancouver Island (Smith, 1939). Individual submarine
valleys are known along the east coast of Korea, along both coasts of Japan and on the
eastern and southern coasts of Formosa. Submarine valleys frequently occur at the
mouths of large rivers, such as the Ganges, the Indus (Fig. 14), the Congo (Fig. 15),
the Ogowe and the Niger. They are also present in different parts of the European and
American mediterranean seas. Some parts of the continental shelf are free from these
canyons, for example the North American coast south of Cape Hatteras or the
eastern coast of Asia south of the Yellow Sea. A summary of the distribution of can-
yons in all oceans and the possible nature of their origin was recently given by
Shepard (1948).

The walls of these subm.arine canyons are usually very steep on both sides, often
with a slope of 5-10° and sometimes 20-35° or even more. These canyon walls
must be made of hard rock since thick layers of soft loose sediments could not be

The Ocean

Fig. 13. Submarine valley off the mouth of the Hudson (according to Smith).

expected to remain at such steep angles for any length of time without collapsing.
Nor can it be supposed that they have been washed out of thick soft bottom sediments
since they would then hardly be permanent. On the other hand, they appear definitely
to be quite young formations that have been formed only in recent times; they appear,
at least in part, to be connected with earthquakes, tectonic breaks and fissures. For a
description of the morphology of these canyons see especially the work of Shepard
and his collaborators (1933, 1938); concerning their probable origin see particularly
Daly (1936) and Kuenen (1938); reference might also be made to the interesting
work of Cooper and Vaux (1949), of Kullenberg (1954), Hecson, Ericson and Ewin
(1954). They have been discussed from the purely geological standpoint by Jessen
(1943), and a survey has been given by Kaehne (1941).

Turning to the general form of the deep sea bottom it is immediately obvious that the
rises and ridges that divide the ocean are features of such enormous size that they
could scarcely occur on the land. The most prominent of these features is the Atlantic
Ridge that extends from Iceland through the Azores, Ascension and Tristan da
Cunha to Bouvet Island and resembles an enormous mountain range 20,000 km

The Ocean

23°40'

23° 20'

67^20 67°40 .

Interval between contours=50fathoms
Fig. 14. Submarine valley off the mouth of the Indus.

E 12° 20'

Tnfervol between contours = 50 tbtfioms
Fig. 15. Submarine valley of the Congo.

The Ocean

long. It divides the Atlantic Ocean into two parts: the eastern and the western At-
lantic troughs. These two elongated depressions are further divided into basins by
transverse ridges. The peculiar relief features of the Atlantic Ridge which forms the
axis of the Atlantic and runs roughly parallel to the continental coast on both sides is
regarded by many as the beginning of a mountain fold, but it could also be the rump
of an old one (Kossmat, 1931).

The Indian Ocean shows a similar division. Here also there is an Indian Ocean
Ridge dividing it into an eastern and a western half, though these two halves appear to
be less subdivided. The Pacific Ocean, on the other hand, is largely a single basin (see
p. 29).

Amongst the most prominent features of the oceanic bottom topography are the
narrow elongated arcs of marginal deeps that lie near the surrounding mountain
chains (or island chains) of the Pacific basin and contain the greatest ocean depths.
These remarkable depressions are confined exclusively to the margins of the Pacific
Ocean; they can also be found in the Sunda arc in the eastern Indian Ocean, in the
Caribbean, in the middle Atlantic Basin and in the south Sandwich marginal deep in
the western part of the South Atlantic. They are usually termed "deep-sea trenches"
or "troughs". This has reference only in a morphological sense and not to its origin.
They are very closely connected with folding processes in the earth's crust, and to some
extent are the counterpart of the mountain chains of the land, and have a related origin.
As an example, the Mariana marginal deep is shown in Fig. 16 both on an isobathic
chart and in a profile perpendicular to its longitudinal extension (Sigematsu, 1933).
Its topographical form is typical of all well-developed marginal deeps. On the side
towards the land, towards the submarine ridge which runs alongside the deep and is
always of mountainous character, the slope is steep, on the ocean side of the deep the
slope is more gentle. On the landward side the angle of the slope may be as much as
20° or more; according to Schott the mean value for a large number of Pacific deeps
is 6-3°. They are always long and narrow.

Table 7. The most important trenches
(With reference to soundings up to 1954)

Greatest

depth (m)

North Pacific Ocean

East Pacific Ocean

Alaska-Aleutian Trench

7679

Chile-Peru J Atacama Trench
Trough \ Arica Trench

7634

6867

West Pacific Ocean

South Mexico /Acapulco Trench

5342

Japan Trench

Trough \ Manzanillo Trench

5121

(Kurillen, Hokkaido, East-Hondo)

10,554

California Trench

4867

Bonin Trench
Mariana Trench
Ryu-Kyu Trench

9156
10,897

7507

East Indian Ocean
Sunda Trough

7455
5664

5257

Philippines Mindanao Trench
Yap Trench

10,497
7141

Andamana Trough

Palau Trench

8138

Atlantic Ocean

Bougainville-New Britain Trench

9140

Puerto Rico Trough

9219

New Hebrides Trench

7570

Cayman Trough

7200

Tonga-Kermandec Trench

10,633

South Sandwich Trench

8264

The Ocean

Fig. 16. The Mariana marginal trench; isobathic chart (lines of equal depth at 1000 m
intervals) and cross-section taken normal to longitudinal axis of the trench.

The Ocean

Table 7 gives a list of the marginal deeps and the greatest depths that have so far been
measured in each. Without doubt these marginal deeps contain the deepest fissures
in the Earth's crust, and in their neighbourhood are the greatest vertical differences in
height that are to be found within a short horizontal distance on the Earth's
crust.

The marginal deeps are conspicuously associated with the volcanic belt which
stretches along the landward side (on island chains or submarine ridges) parallel
with the line of deep-sea trenches and with the earthquake belt which is also present
in the immediate neighbourhood of the trenches, especially on the landward side. This
connection with seismic and volcanic activity is always present and indicates a causa-
tive connection between these phenomena. Another phenomenon closely associated
with the marginal deeps is the strong negative gravitational anomaly occurring along
a very narrow line. The investigations of Vening-Meinesz (1932, 1934) on the gravi-
tational field in the East Indies and later the investigation of Hess (1938) in the West
Indies have clarified this connection. The belt of abnormal gravity does not coincide
exactly with the line of deep-sea trenches, but is displaced towards the adjacent moun-
tain ridge. There exists in all cases a parallelism with the deep-sea trenches, but the
relationship to the topography is more complicated than this. In the Philippine trench
the line of negative anomaly lies directly underneath the trench (see Fig. 17) but it is

800

Fig. 17. Gravity profile over the Philippine Trench at Surigao (isostatic anomaly; observed

values indicated by black dots; the bottom profile shown schematically with a vertical

enlargement by 1:15) (according to Vening-Meinesz).

weak, although the trench is particularly deep; in the Java trench the gravitation
anomaly is very pronounced but lies at the side of the trench (Fig. 18). Since a line of
negative gravitational anomaly is present wherever there is a deep-sea trench, there
must undoubtedly be some connection between the two phenomena. This is also indi-
cated by the relationship of seismic activity and the distribution of volcanoes mentioned
above. For the explanation of this relationship, see especially Vening-Meinesz
(1940).

In addition to the deep-sea trenches there are also the differently shaped, nearly
circular depressions. It cannot yet be decided whether these should be regarded as
deformed marginal deeps but those between the Sunda Islands, the Moluccas, and the
Philippines (Celebes, Sulu, Banda and other deeps) occur in close connection with the

The Ocean

East Indian negative gravitational anomaly. There are similar shaped deeps in the
European Mediterranean, in the Gulf of Mexico and in other places, though not of
the same depth or extent. Amongst these may be reckoned the comparatively small but
very deep Romanche deep which divides the mid-Atlantic Ridge in two, at about
18-19° W. on the equator. The corresponding lowering of the mid- Atlantic ridge
is as low as 4500-4800 m. The great significance of this deep connection between the
eastern and the western troughs for the hydrographic structure of the water masses of

-100

Isost Anomaly

600

Fig. 18. Gravity profile from Benkulen (Sumatra) towards the Indian Ocean (see Fig. 17).

the eastern trough will be discussed later (see Chap. Ill, 5, b). The greatest depth
measured in the Romanche deep is 7230 m. A bathymetric chart of the area has
been given by Stocks and Wtisx (1935).

While the slope of the deep-sea bottom is in general slight and only reaches larger
values at the continental slope, occasionally very steep gradients occur near islands,
submarine banks and reefs. As on land there has often been major volcanic activity on
the sea bottom, partly in extended zones associated with the deep-sea trenches and
partly more widely spread. The steepest slopes are always those of the purely oceanic
islands which are all of volcanic origin; these slopes are of the same order of magnitude
as those of land volcanoes. The slope of the island St Helena, for example, over short
distances is as much as 38-40° and the Atlantic island St Paul has slopes of 62°.

In numerous cases the volcanic forces have been insufficient to build an island cone
up to the surface. They form submarine peaks, whose summits may still be some
hundreds of metres below the surface and seldom come up to normal anchorage
depths. These submarine volcanic cones were only occasionally found by wire sound-
ing, which allows them to be quickly and accurately charted. In this connection there
might be mentioned the surveys of the area of the Bogoslov volcano (Bering Sea) by
the United States Coast and Geodetic Survey (Smith, 1937) and the survey of the
"Altair" peak (Defant, 1939).

4. Arrangement of the General Bottom Topography of the Individual Oceans

For an elucidation and abbreviation of the following discussion Plate 2 is presented,
and it shows all the main characteristic features of sea-bottom topography in a clear
manner. For each ocean there is a list of the principal features which have been desig-
nated by letters and numbers on the plate. The capital letters show the deep-sea basins
(troughs) in succession for each ocean, the small letters denote the ridges and rises
that separate these basins, and the numbers indicate the deep-sea trenches.

Im Ocean

c Ocean

Deep-sea basins

Ridges and rises

North America Basin

North and South Atlantic Ridge

Brazil Basin

Rio Grande Rise

Argentina Basin

Whalefish Ridge

Cape Verde Basin

Atlantic Indian Ridge

Sierra Leone Basin

Guinea Rise

Guinea Basin

Sierra Leone Rise

Angola Basin

Cape Basin

Deep-sea trenches and troughs

Agulhas Basin

Cayman Trough

Atlantic-Antarctic Basin

Puerto Rico Trough

South Antilles Basin

South Sandwich Trench

Romanche Trench

The topography of the Atlantic Ocean bottom is characterized by its division into
East and West Atlantic Troughs by the Atlantic Ridge. This ridge begins at Iceland ;
from the Iceland shelf it runs south-westward as the narrow Reykjanaes Ridge whose
bottom form was fixed by the soundings of the "Meteor" (Bathymetric chart by
Defant, 1930, 1931, 1936). At 5 1 ° N. the ridge broadens out somewhat towards the
west (Telegraph Plateau), and then runs into the Azores Plateau which can be regarded
as a great extension of the central ridge to the east and south-east. The ridge then
narrows and remains at a depth of 2500-3500 m and apart from St Paul Island
supports no islands as far as the equator. At 7-8° N. 36° W. there is a gap which
reaches to a depth of 4400 m. The greatest gap is, however, on the equator near
the Romanche Trench (see p. 27). South of this the ridge is broad and rounded and
carries the islands Ascension (height 860 m), Tristan da Cunha (2329 m), Gough
(1335 m) and Bouvet (935 m). St. Helena belongs to a minor ridge farther to the east.
These extended minor ridges are peculiar to the section of the ridge between 0° and
20 "" S. The South Atlantic Ridge is connected west of Bouvet Island by the Atlantic
Indian Ridge to the Crozet and Kerguelen Ridges of the Indian Ocean. The Atlantic
Ridge extends over 20,300 km and is by far the longest underwater mountain system
on the Earth.

The Eastern and the Western Atlantic Basins are further divided by transverse
ridges. An outline of the main division is shown in Plate 2 where the geographical
arrangement of the basins is particularly clearly shown. A special characteristic of the
Atlantic Ocean is that it is completely closed in the north towards the Arctic Sea and
the Norwegian Sea below a depth of about 500 m. This has far-reaching oceano-
graphic consequences. In contrast to this nearly complete blocking of the deeper layers
to the north, the Atlantic in the south is completely open down to great depths to the
Atlantic-Antarctic Basin.

There are topographical differences between the eastern and the western troughs
that have a considerable effect on the oceanographic structure. The transverse ridges
are not as well developed in the western trough as in the eastern, and particularly the
Rio Grande Ridge, which is somewhat better developed, has deep openings that per-
mit continuous communication from the Atlantic-Antarctic Basin through the Ar-
gentina Basin, the Brazil Basin and the Guiana Basin to the North America Basin
below 4000 m. In the eastern trough, on the other hand, the Whalefish Ridge, which

The Ocean

separates the Cape Basin from the Angola Basin, forms a continuous diagonal
transverse barrier. It rises steeply from a depth of 5000-5500 m to only 964 m,
forming an unbroken submarine wall connecting the mid-Atlantic Ridge between
Tristan da Cunha and Gough Island with the broad shelf of the African mainland. All
the other ridges in the east Atlantic trough have openings that reach below 4000 m.

{b) Indian Ocean

Deep-sea Basins

Ridges and rises

Arabian Basin

Bengal Ridge

Somali Basin

Carlsberg Ridge

Madagascar Basin

Diego Garcia Bank

Agulhas Basin

Central Indian Ridge

South-westlndian Antarctic Basin

Mascarene Radge

South-east Indian Antarctic Basin f

Atlantic-Indian transverse Ridpe

South Australian Basin

Crozet Ridge

India-Australia Basin
Deep-sea trendies

h
i

Kerguelen-Gaussberg Ridge
Macquarie Ridge

Sunda Trench

Nicobar Trench

It is only in more recent times that it has been found that the Indian Ocean is also
divided into two large troughs by a central ridge. This central ridge runs north-
westward from the Kerguelen-Gaussberg Ridge, gradually narrowing, then through
the elevation around the volcanic islands of New Amsterdam and St Paul in the section
between the 20° and 0°, where it reaches its highest elevation. Here it carries the shallow
waters and banks of the Saya da Malha and the Nazareth Bank. Two outlying ridges
run out from this point, one to the north-west to the Seychelles and the Amirantes,
and the other to the south-west, here it carries the islands of Mauritius and Reunion.
In this middle section the ridge stretches over more than 10° of latitude. From here it
splits into two parts running towards the north. The eastern part carries the Chagos
Islands and runs up through the Maldives and the Laccadives, gaining a connection
to the south-west Indian shelf. The western part, which was first mapped by the
Danish "Dana" Expedition (Carlsberg Ridge), is much narrower and not as high.
This Indian Ridge is also of enormous length and runs from the South Arabian Sea
to the edge of Antarctica at Kaiser Wilhelm Land (WusT, 1934).

Deep-sea basins

Deep-sea trendies

Central Pacific Basin

Aleutian Trench

Philippines Basin

Kurile Trench

Caroline Basin

Japan Trench

Coral Basin

Bonin Trench

Fiji Basin

Mariana Trench

Tasman Basin

Japan Trench

South Pacific Basin

Philippines Trench

Berlinghausen Basin

Riukiu Trench

Peru-Chile Basin

Bougainville-New Britain Trench

Califomian Basin

New Hebrides Trench

Banda Sea

Tonga Trench

Celebes Sea

Kermadec Trench

North China Sea

Chile Trench

Peru (Atacama) Trench

Califomian Trench

30 The Ocean

Ridges and rises

a Bonin Ridge

b Eastern Pacific longitudinal Ridge

c South Pacific transverse Ridge

d Macquarie Ridge

e Fanning Ridge

f Hawaii Ridge

g Fiji Ridge

h New Hebrides Ridge

As has already been mentioned above (see p. 24), the deep-sea trenches that are a
major characteristic of the Pacific Ocean are marginal, that is, they occur around
the rim of the ocean, either near the coast or beside outlying island chains. The main
part of the ocean forms a vast deep-sea basin that, judged by the rather sparse sound-
ings available, is not as strongly subdivided as the Atlantic and the Indian Oceans.
The western, and especially the north-western open Pacific Ocean, contains the greatest
continuous extension of the sea bottom below 5000 m and wide areas have a depth
even greater than 6000 m. The eastern and south-eastern parts are less deep. Sound-
ings have confirmed the deep-sea division, apparent from the individual chains of
islands, along a direction from north-west to south-east. In the central part of the
ocean, especially to the south, there are groups of islands that are not associated
with deep-sea trenches and that occur in clusters. It was earlier supposed that these
were on top of plateaus or ridges at no great depths. More recent soundings have
shown, however, that this is not the case; only islands that are very close have any
submarine connection, and the others usually rise separately as volcanic cones from
very great depths and form a very characteristic topographical feature of the South
Pacific.

{d) Mediterranean and Adjacent Seas

The Atlantic Ocean is connected with the greatest number of mediterranean seas,
which have also greatest extent. These are the Arctic Sea, which can also be regarded as
a continuation of the open ocean across the Greenland-Iceland-Faroes Ridge, and
the American and European mediterranean seas.

The North Polar Sea, also known as the Arctic Mediterranean, includes: (1) the
North Polar Basin surrounded by the seas of the flat shelf of Northern Europe and
Northern Asia (Barents Sea, Karelian Sea, West Siberian Sea, Nordenskjold Sea, the
East Siberian Sea and the Tjuktjen Sea) and of North America (Beaufort Sea and the
large number of sea straits in the North American archipelago); (2) the European
North Sea south of the Spitzbergen Ridge (depth 1750 m); and (3) the Baffin Sea.
The total area amounts to 14-06 million km^.

The European North Sea is divided by a ridge at a depth of about 2400 m, running
from Iceland through Jan Mayen to the Bear island into two basins; the southern
Norwegian deep and the northern Greenland deep, both with a depth of over 3000 m.
For the bottom topography of the North Polar Basin see WiJST (1941).

The American Mediterranean is divided by the coastal orography and by the bottom
topography into three areas: the Mexico Basin (1-602 million km^), the Yucatan
Basin (0-760 million km^), and the Caribbean Basin (1-948 million km^) with a total
area of 4-310 million km^, A new bathymetric chart has been prepared by Stocks

The Ocean 31

(1938) taking into account numerous recent soundings. The Caribbean Basin is itself
further subdivided by two north-south ridges the Beata and the Aves Ridges into
three parts : the Magdalena Basin in the west, the Venezuela Basin in the middle and
the Aves Basin in the east.

The general form of the bottom topography of the whole of the American medi-
terranean basins shows considerable regional differences that can be explained by
their different origins (see Dietrich, 1937, 1939). All three basins are to a large extent
cut off from the Atlantic Ocean; this is of decisive importance for the question of
renewal of the deep water of the individual basins. The Gulf of Mexico is connected
with the free ocean only through the Florida Straits (sill depth 800 m) and with the
Yucatan Basin through the Yucatan Channel (sill depth 1600 m). The Yucatan Basin
and the Caribbean Sea are connected over the Jamaica Ridge with a sill depth of not
more than 1400 m. The Yucatan Basin has a single connection with the Atlantic
Ocean, the Windward Passage between Haiti and Cuba with a sill depth of about
1600 m. The Caribbean Sea is connected with the open ocean by several gaps between
the West Indian Islands, the deepest of these are the Mona, the Jungfern and the
Anegada Passages, which are the only ones concerned in the renewal of the deep
water of this mediterranean sea. Their sill depths are 1600-1620 m and 1780-1800 m,
respectively.

The European Mediterranean Sea. This falls into two clearly separated main divi-
sions, the Western Mediterranean from the Straits of Gibraltar (sill depth 320 m)
to the Sicilian Ridge (sill depth 324 m), and the Eastern Mediterranean. To the
latter are connected the Adriatic Sea and the Aegean Sea which in turn is connected
through the Dardanelles (sill depth 57 m) with the Sea of Marmora and further,
through the Bosphorus (sill depth 37 m) with the Black Sea, A modern bathymetric
chart for the European Mediterranean has been given by Stocks (1938). The Western
Mediterranean is separated by a ridge running from Tunis through Sardinia, Corsica
and Elba to the Italian mainland into two basins: the Balearic Basin in the west and
the Tyrrhenian Basin to the east (greatest depth 3731 m). The Eastern Mediterranean
goes down to considerable depths (more than 4000 m) especially in the Ionian Basin ;
the greatest depth is 4715 m south-west of Cape Matapan.

Of the smaller mediterranean seas around the Atlantic, the Baltic and the Hudson
Bay may be mentioned, but will not be described further since they have largely the
character of shelf seas. The mediterranean seas of the other oceans are also of the
same type except for the Red Sea which is an elongated canyon-like trough with depths
of more than 2000 m and forming a real trench between the coastal strips of the
Arabian and Egyptian plateaus. Its outlet in the south is the Strait of Bab el Mandeb
with a sill depth of about 150 m. The Persian Gulf is a shelf sea with depth less
than 100 m (Stocks. 1944).

Chapter II

The Sea- water and its Physical and
Chemical Properties

1. Collecting Oceanographic Samples

The ocean basins are filled with a liquid that is essentially the same as rain water
formed by the condensation of water vapour. An accurate knowledge of the different
contents of sea-water is indispensable in order to be able to learn something of the
geophysical-chemical structure of the ocean. This knowledge of the structure must be
derived from samples collected at oceanographic stations. It cannot be limited to the
surface layers of the sea but must include all layers down to the sea bottom and must
be based on a network of observation stations placed as systematically as possible.
The precise determination of the spatial distribution of the oceanographic factors is a
major achievement of modem oceanography and its observational technique.

Collecting samples from the surface of the sea offers no real difficulties, or at the
most only those that can be overcome by simple means. The collection of unob-
jectionable and homogeneous material of definite origin from deep layers of the sea
is, however, not easy and it has required the work of several decades to overcome the
difficulties. The differences in the oceanographic factors (such as temperature and
salinity) at deeper levels become continuously smaller both in horizontal and vertical
direction; the accuracy of measurements at great depths must therefore be increased,
and it has only been possible by the use of modern analytical techniques to do this
with the degree of accuracy needed to follow small local variations.

Almost all the properties of sea-water, apart from the temperature, can be deter-
mined if genuine samples of water are available from each particular depth, because
these properties show no appreciable alteration when the sample is brought from the
deep sea to the surface. The temperature of the water must, however, be determined at
the place and at the depth from which the water sample was taken {in situ).

To collect oceanographic data at a station it is necessary to lower a thermometer in
order to measure the temperature at different depths, and to bring back genuine samples
of water from these depths in sampling bottles. The work at such an oceanographic
station is done with a series-machine so-called because it is usually used for series
observations, that is, the sampling bottles and thermometers are lowered at the same
time to predetermined depths and a series of samples is collected and brought back
together with temperature measurements. More recently, specially built machines
have been used for this, but sounding winches or hydrographic winches were used
previously. The oceanographic series machine and its operation on board ship will not
be described here, but details are given "'Meteor'" Work, 4, No. 1 (WiisT, Bohnecke
and Meyer, 1932, Berlin).

The Sea-water and its Physical and Chemical Properties

i ' 'Nil

Before
turning

After

turning

Fig. 19. Water bottle used on the "Meteor" Expedition and method of operation.

34 The Sea-water ami its Physical and Chemical Properties

Sampling bottles and thermometers are the most important of the instruments used
at an oceanographic station. To be suitable for series observations the sampling bottle
must be as light as possible; while still having sufficient capacity, it must allow free
circulation of water and it must function and close reliably. There are many differen
models of sampling bottles. They are all lowered open, allowing the water to pass
through freely as the bottle sinks and are closed automatically for hauling to the sur-
face. The most successful design is that of Nansen with two plug valves. The series
water bottle used by the "Meteor" Expedition 1925-27 was constructed on the same
principles but was a little larger and had a number of minor improvements. This water
bottle and its function is illustrated in Fig. 19 (WiJST, 1932). It had a capacity of
1250 cm^, weighed 44 kg (with thermometer frame approx. 5 kg) and had an over-
all length of 75 cm. Among the older designs may be mentioned that of Ekman
(1905) with improvements by Knudsen (1923) and a special 4 1. water bottle {'"Meteor"
Report, 4, No. 1, 1932).

Small 100-200 cm^ bottles of ordinary green glass are suitable for storage of
water samples (for chlorine titration and analysis) since they have been found by the
investigations of Helland-Hansen and Nansen to have very slight solubility; they are
fitted with a patent stopper with a porcelain head carrying the sample number.
Before use the bottles must be boiled, cleaned with chromic acid-sulphuric mixture,
rinsed with distilled water and very carefully dried.

A definitive programme has been worked out for the work required at each oceano-
graphic station and this has been found to be very successful as, for instance, during
the "Meteor" Expedition 1925-27, and has been described in Vol. 4, No. 1 of the
''Meteor'' Report. It is worth mentioning particularly that a machine and an obser-
vations schedule containing everything of importance in the working programme for
the series should be kept for each oceanographic station. Very often the results of an
oceanographic series depend on the careful compilation of the machine and observa-
tions schedules. Apparently unimportant details may become important later during the
interpretation of the observations and can contribute to the uniformity and homo-
genity of the observations.

2. Temperature Determination for all Layers of the Ocean

The determination of the temperature of the surface layer of the sea offers little
difficulty. A sample taken from water collected in an ordinary bucket, lowered into
the sea for a short lime while the vessel is under way, is put immediately in a shady
place and its temperature is taken with a sensitive thermometer while at the same time
it is kept stirred. The water sample must be drawn from as far forward as possible (on
steam ships forward of the condenser exhaust). See Lumby (1927) on the measurement
of surface temperatures and the collection of suitable water samples. New surface
sampling bottles have been designed by Sund (1931) and improved by Schumacher
(1938).

The determination of the temperature of the deeper layers of the sea is considerably
more difficult, and this also needed the work of almost a decade to reach an accuracy
suitable for scientific requirements. In the upper layers temperatures correct to 0-1 °C
are usually sufficient, but in the deep layers the variations both horizontally and ver-
tically are usually so small that an accuracy of 0-01 °C is needed to get some idea of

The Sea-water and its Physical and Chemical Properties 35

the spatial variations in temperature. This accuracy is also necessary for the calculation
of densities accurate to the fifth decimal place. Deep-sea thermometers are thus
extremely accurate and sensitive instruments which cannot be handled skilfully just
by anyone.

An ordinary thermometer suspended freely in the water will not show the correct
temperature since the pressure of the water will compress the thermometer bulb and
force the mercury to a higher level. It is therefore necessary to protect the thermometer
against the water pressure by enclosing it in a thick-walled glass tube. The part of the
tube surrounding the thermometer bulb is filled with mercury to improve the heat
transfer between the water and the bulb. Since the temperature usually decreases with
depth the instrument first used was a maximum and minimum thermometer con-
structed by Six and adapted for deep-sea use, and this was the classical instrument
used on the "Challenger" and the "Gazelle" Expeditions. Since 1874 the reversing
thermometer, first produced commercially by the firm Negretti and Zambra, has been
used instead, and with numerous modifications is still used at the present time as the
standard instrument for oceanographic temperature recording. This is a thermo-
meter with the capillary considerably constricted a little above the mercury bulb,
so that the mercury thread will break at this point when the thermometer is turned
through 180° and slide down to the other end of the capillary. The higher the tem-
perature when the thermometer is reversed the longer the mercury thread that is
broken off. This thread gives a direct reading of the temperature at that time when
read against a scale running in the reverse direction with appropriate corrections. The
accuracy of the thermometer is very dependent on the shape of the constriction. It
must, of course, be made so that the mercury thread always breaks at the same point
and it must be designed so that further mercury cannot follow the thread if the ther-
mometer passes subsequently through a warmer layer of water. All the initial diffi-
culties were overcome by the work of Richter (of the firm Richter & Wiese, BerHn) so
that the reversing thermometer is now a true precision instrument. The shape of the
constricted part of the capillary is shown in Fig. 20. Further details are given in the
''Meteor'' Report, 4, No I, by Bohnecke (1932), and in "Oceanographic Instrumenta-
tion" (Rep. Conf. Rancho Santa Fe, Calif. 21-23 June 1952, p. 55).

In use the reversing thermometer is enclosed in a suitable holder (a brass tube)
which is attached directly to a reversing sampling bottle or to a frame which can be
reversed at the desired depth (reversing frame, propeller frame).

The reversing thermometer does not show the true temperature {in situ) directly
since it will have been brought back to the surface through layers of water at different
temperatures. After removal from the sampling bottle on deck it is placed immediately
in a water bath and allowed to adjust to the water temperature before it is read. To
show the temperature of the water bath every reversing thermometer is fitted with a
normal auxiliary thermometer. To correct the reading to the temperature in situ a
small correction given by the formula

{T -t){r+ Kq)
6100

J ^ {r~t){r+ V,)

6100

must be applied. In this equation T' is the uncorrected reading of the reversing ther-
mometer, t the reading of the auxiliary thermometer (bath temperature), Vq is the

The Sea-water and its Physical and Chemical Properties

Fig. 20. Reversing thermometer (with visible constriction).

volume of the small bulb and the capillary of the main thermometer until 0°C and
expressed in degree units on the capillary scale, 1/6100 = jS being the coefficient of
expansion of mercury. The corrections given by the formula are listed in tables to
allow quick accurate working (Schumacher, 1923, 1933; Hidaka, 1933; Geissler,
1934). Kalle (1953) has given a simple graphical method for the determination of the
corrections (C). A calibration correction has to be added to the corrected reading
of the thermometer.

By very careful attention to all the factors involved (continual checking of the re-
versing apparatus, accurate readings using a magnifying glass, checking the zero
point, proper correction) the mean error in the temperature determination can be
kept down to, on the average, ±0-01 °C. This method gives the temperature at single
points in the ocean and is of considerable use in series observations at oceanographic
stations. For a special purpose, however, it may be desirable to have a continuous
record of the temperature at a fixed depth or to obtain quick successive readings of the
temperature in a particular layer. A thermograph is usually used for the first purpose
(at coastal stations or for continuous recording of the temperature at the surface of
the sea from a moving vessel). For greater depths diff'erent types of electrical resistance
thermometers have been designed but they have not yet proved very satisfactory in
use. For a rapid survey of the upper 150 m of the sea or for a continuous registra-
tion of the vertical temperature gradient of this upper layer to about 200 m, Spil-
HAUS (1938, 1940) has developed and tested a bathythermograph. This has proved
successful and offers considerable advantages where rapid changes of temperature
can be expected. For greater depths Mosby (1940) has designed a "thermosounder"
that has given useful results.

3. Salinity and its Determination

One of the most important properties of water is its ability to dissolve a very large
number of solids and gases without chemically reacting with them. As a consequence

The Sea-water and its Physical and Chemical Properties

of this property all the water on the earth is more or less impure, that is it contains in
addition to chemically Hnked hydrogen and oxygen (HgO) a number of other substances
in varying amounts. If the salt content, the salinity, were defined as the weight of all
the salts dissolved in a kg of sea-water this would provide to be the simplest numerical
specification of the amount of dissolved salts in the water. Unfortunately it is rather
difficult to measure this definite quantity since, when sea-water is evaporated to dry-
ness and heated to red heat to remove the last traces of water, some hydrogen chloride,
carbon dioxide and a small amount of hydrogen bromide are also lost. This loss is not
easily compensated with sufficient accuracy by adding a corresponding correction.
At the suggestion of Forch, Sorensen and Knudsen (1902) the salinity has been de-
fined as the total amount of solid material in grammes contained in 1 kg of sea-water
when all the bromine and iodine have been replaced by the equivalent amount of
chlorine, all the carbonate converted to oxide and all organic matter has been com-
pletely oxidized. The salinity defined in this way can be determined with great accuracy
and can thus serve as a basis for the investigation of the relationship between any
single component and the total salinity.

Sea-water is a dilute solution of a mixture of salts; in such an aqueous solution salts,
acids and bases are more or less completely electrolytically dissociated (Arrhenius
and van't Hoff). The chemical compounds precipitated on evaporation of such solu-
tion are in solution split into atoms or groups of atoms with an electric charge, either
positive (cations) or negative (anions). The electrical charges balance exactly so that
the solution remains electrically neutral. The constituents of this mixture of salts
are therefore listed as their ions. Table 8 shows the composition of a typical sample of
sea-water with a salinity of 34-40%o.

Table 8. The principal constituents of sea-water
(34-40 /oo salinity)

Cations

Sodium

Potassium

Magnesium

Calcium

Strontium

g/kg 1 mmole/kg

10-47
0-38
1-28
0-41
0-013

455-0

9-7

52-5

10-2

0-15

percent-
age of S

30-4
1-1
3-7
1-2
0-05

Anions

Chloride

Bromide

Sulphate

Bicarbonate

Borate

g/kg

18-97
0065
2-65
0-14
0027

mmole/kg

5351
0-81

27-6
2-35
0-44

percent-
age of S

55-2
0-2
7-7
0-4
0-08

It was formerly customary to give the constituents of sea-water in terms of the com-
pounds that were precipitated on evaporation. Dittmar (1884) has given the figures
shown in Table 9 as the mean of seventy-seven very complete analyses of sea-water
samples made by the "Challenger" Expedition; they have been calculated on the basis
of a salinity of 35 g of salts in 1 kg of sea-water.

In the open ocean the total concentration of salinity varies between moderate
limits, usually between about 33 and 38%o depending in the first place on the climate
(precipitation, evaporation and in polar regions ice melting). In coastal areas where
there is a considerable inflow of fresh water from rivers and from ground water the
salinity may have a considerably lower value. Especially in the almost closed adjacent
seas of higher latitudes (such as the Baltic) with low evaporation, a considerable

The Sea-water and its Physical and Chemical Properties

Table 9. The salts obtained from sea-water
(Calculated as 35 g of salts per kg)

Salt

weight in g/kg
sea-water

Percentage of
total salts

Sodium chloride OJaCl)

27-213

77-758

Magnesium chloride (MgCl)

3-807

10-878

Magnesium sulphate (MgSO^)

1-658

4-737

Calcium sulphate (CaS04)

1-260

3-600

Potassium sulphate (K2SO4)

0-863

2-465

Calcium carbonatef (CaCOg)

0123

0-345

Magnesium bromide (MgBrg)

0076

0-217

Total

35000

100000

t Includes all the other salts present in trace amounts.

inflow of fresh water and precipitation on the surface may have a low saHnity (8-
5%o) and at the inner ends mostly only brackish water with 1%^ or even lower. The
highest salinities are to be found, on the other hand, in the subtropical adjacent seas
with almost no inflow of fresh water, no precipitation and strong evaporation as, for
instance, in the Red Sea and in the Persian Gulf which at the inner ends have maximum
salinities of almost 40%o.

While the salinity is always liable to show some variations the proportion of the
different ions in sea-water is remarkably constant. This constancy which is of con-
siderable oceanographic importance is only further confirmed by all carefully made
analyses. Accurate chemical analysis of the samples collected by the "Challenger"
Expedition from almost all parts, and depths of the ocean demonstrated this constant
proportion between the individual constituents and more recent investigations as
shown in Table 10 have led to the same results.

Table 10. Analysis of the salt content of sea-
water (percentages)
(DiTTMAR, 1884; Makin 1898; Wheeler, 1910)

No. of samples

55-29

55-18

55-29

0-19

0-13

—

SO4

7-69

7-91

7-56

CO3

0-31

0-21

0-37

30-59

30-26

30-76

1-20

1-24

1-22

3-72

3-90

3-70

The mean ratio Mg : CI is 0-0682 and for SO4 : CI the ratio is 0-1397. The most
recent analyses by Matthews, Thompson, and others, have given a value of 0-6802
(with limits of 0-6785 and 0-6814) for the first ratio and 0-1395 (with limits of 0-1387
and 0-1403) for the second. Using very accurate analyses calcium and bicarbonate

77?^ Sea-water and its Physical and Chemical Properties 39

show sometimes smaller variations from the above-mentioned general propor-
tionality (not more than 1%) which are due to biological processes (precipitation of
calcium carbonate), to the solution of calcium carbonate from sea bottom and in
coastal areas to the inflow of river water (containing calcium carbonate).

The very constant proportions of the ions present in sea-water allow chlorine to be
used as a measure of the salinity of a sample of sea-water. This was done many
years ago by Forchhammer (1859, 1865) and later by Knudsen (1902), from a very
careful examination between 2-69 and 40-18%o, derived the simple equation

S = 0-030 + 1-8050 CI,

which is now used generally for the calculation of the salinity (S) from the chlorine
content. This salinity is that given in the definition above. It is a little smaller than the
actual salt content (by about 0-14%o) but since it is the differences in salinity that are
important this has very little significance.

The most convenient method for the determination of salinity is that of Mohr
(1956) in which the sample is titrated with silver nitrate with a calcium chromate
solution as indicator; this is also suitable for use on board ship. This chemical method
gives a relatively fast and accurate determination of the chlorine in sea water, and the
salinity can be calculated from this value using the equation given above. This method
is the usual method used at the present time in practical oceanography (see especially
Meyer (1932) for the practical details of the titration and the necessary working rou-
tine).

The chlorine titration is only a relative determination, and to find the absolute value
it is necessary to standardize the solution used for titration against the "Normal
water" introduced by Knudsen (1903, 1925); this standardization very largely elimi-
nates the effect of the subjective assessment of the colour of the indicator. Normal
water is sea-water kept in sealed glass tubes of which the chlorine content has been
very accurately determined, formerly by the central laboratory of the International
Hydrographic Institute in Copenhagen, and at the present time by the Woods Hole
Oceanographic Institution. The difference between the value obtained by titration
of the normal water and that marked on the tube gives the total error in the titration.
Knudsen (1901) has prepared hydrographic tables for the comparison of chlorine
determinations of sea-water with different salinities with the chlorine determination
made on normal water.

If the average salinity of the ocean is taken as 35%o then calculation gives the total
amount of salt in the ocean as 4-84 x lO^*' tons; this corresponds to a volume of 21-8
miUion km^ which, spread evenly over the sea (361 million km-), would be a layer of
salt 60 m thick.

In addition to the substances already mentioned, sea-water also contains traces of
a large number of elements which are of little importance for oceanography, though
they are probably important in the metabolism of marine organisms. The determination
of the concentrations of these elements presents very great analytical difiiculties and
the older determinations must be treated with great caution. Table 1 1 shows a more
recent list of the elements present in the sea according to Kalle (1945), which is
based on a similar one given earlier by Watterberg (1938). In many cases the figures
given represent only the order of magnitude of the concentration of an element. Of

The Sea-water and ifs Physical and Chemical Properties

the elements that are present in somewhat greater concentration may be mentioned
iron, copper and gold. Iron is present in extremely small quantities and sea-water
is probably one of the naturally occurring materials poorest in iron. The importance
of copper can be seen from its occurrence in place of iron in the blood pigments of
many marine animals (hccmocyanin). The occurrence of gold in sea-water at one
time aroused particular interest since, according to older determinations, the isola-
tion of gold from sea-water was technically promising. These older determinations
have, however, been shown by the results of Haber (1928) and Jaenicke (1935) to
be incorrect, and the gold found came largely from the reagents used, from the air and
from the glass of the apparatus. The gold content of sea-water found by analysis of
the samples collected on the "Meteor" Expedition was only 4 x lO"'* g/kg of sea-
water, a concentration which would be of no technical use.

Table J I. Concentrations of the trace elements present in sea-
water in milligrams per cubic metre
(According to Kalle, 1945)

Fluorine

1400

Selenium

Silica

1000

Uranium

Nitrogen (NO", NO;, NH3)

1000

Caesium

Rubidium

200

Molybdenum

0-7

Aluminium

120

Cerium

0-4

Lithium

Thorium

0-4

Phosphorus

Vanadium

0-3

Barium

Yttrium

0-3

Iron

50(2)

Lanthanum

0-3

Iodine

Silver

0-3

Arsenic

Nickel

Copper

Scandium

004

Manganese

Mercury

003

Zinc

Gold

0004

Radium

00000001

The radioactivity of sea-water has been accurately investigated in recent times, and
detailed examinations have been made principally by Pettersson (1937, 1938), and
Thompson and his collaborators (1932). According to these investigations the radium
content of sea-water with a salinity of 35%o varied between 0-04 and 0-2 x IQ-^^ o/^
(or between 0-04 and 0-2 billionth parts of a gramme per litre); 0-07 x 10-^^ 0/^
radium can be taken as mean value. Deep water has a uranium content of 1-5-2 x
10"'' %o; surface water has a somewhat lower value. The thorium content is less
than 0-5 x 10-« %«.

Since the radium content of sea-water is 10-000 times less than that of rocks of the
Earth crust, it is extremely small and corresponds to only 10% of the amount that
would be in equilibrium with the uranium content. According to the view of Petters-
son, this remarkable deficiency of radium in the sea can be attributed to the very
rapid precipitation of the iron carried into the sea, almost entirely as ferric hydroxide.
In the precipitation the thorium and its isotope ionium that immediately precedes
radium in the disintegration series are co-precipitated. The ionium produced from the
uranium in solution in the sea is thus steadily removed by precipitation of the iron.

The Sea-water and its Physical and Chemical Properties 41

Only that part of the element remaining in solution disintegrates to give radium and its
disintegration products in the sea (see also Hess, 1918).

4. The Density of Sea-water and its Dependence on Temperature, Salinity and Pressure

The density p of a material is the mass of a unit volume [g cm"^]. Frequently the
specific weight is given instead of the density; this is defined as the quotient of two
densities p/p„., where p is the density of the substance in question and p,,, is the density
of distilled water at a fixed temperature. The specific weight is thus a dimensionless
quantity. In the CGS system the density and the specific gravity are numerically
equal if distilled water at 4°C is taken as the comparison liquid.

Due to its salt content sea-water is heavier (more dense) than pure water. The den-
sity is always fairly close to 1 and varies depending on the salinity S, the temperature
/ and the pressure p between narrow limits ; for example, at the surface of the open
ocean between 1-02750 and 1-02100. For oceanographic purposes it is necessary to
know the density correctly to at least 5 decimal places. For simplicity instead of using
p it is customary to use a density value o derived from the equation a = (p— 1) x 10^;
for instance instead of p = 1-02754, a = 27-54 is used. Very often the reciprocal of
the density 1/p = v, the specific volume [cm^ g-^] is used. This also is required cor-
rect to the fifth decimal place and for simplicity and convenience only the last three
figures are given according to the equation a = (i; — 0-97) x 10^. For example when
V = 0-97320, a = 320.

The dependence of the density and the specific volume on the temperature, the
salinity and the pressure were first investigated at the beginning of this century (1899)
by an international commission headed by K>ajDSEN (1902, 1903). The relationship
of the density at 0°C and atmospheric pressure at sea-level to the chlorinity is given
by

ao = -0-069 + 1-4708 CI - 0-001570 Cl^ + 00000398 C\\

This equation is valid for chlorinities between 1-47362 and 22-2306.

The dependence of the density of sea-water on the temperature requires a knowledge
of the thermal expansion of sea-water. The thermal expansion coeflficient determined
in the laboratory shows that the density has a pronounced dependence on the tem-
perature; at atmospheric pressure (sea surface) is given by o-^ — CTq — Z). Z) is a
very complicated function of a^ and of the temperature / and has been given to the
fifth place in Knudsen's hydrographic tables (1901). Schumacher (1922) has also given
graphical tables, and further tables for the determination of the density of sea-water
under normal pressure have been given by Matthews (1932) and Thorade and Kalle
(1940). These tables show that an increase of 0-01%o in the salinity gives an approxi-
mate increase in the density (ct^) of 8 units in the third decimal place. The increase is
about the same for all temperatures and salinities. For low and high temperatures
the density change is very different and depends also somewhat on the salinity.
Figure 21 (Helland-Hansen, 1911-12) shows the eff"ect of variations in temperature
on the densities of distilled water and of sea-water with sahnity 35%o. From the re-
lationship between temperature and density the temperature of maximum density
can be determined for different salinities. This is also given with somewhat less ac-
curacy by the equation

/max = 3-95 - 0-266ao.

The Sea-water and its Physical and Chemical Properties

-3
-2

Seowater5 = 357oo

■^

^Pure water

-2 0

12 16
t, "C

20 24 28

Fig. 21. Effect of changes in temperature on the density of pure water and of sea water at

35^00 salinity.

Thus for different salinities, where

S in %„ = 0 10 20

cmax = 000 818 16-07

/max in °C = 3-947 1-860 -0-310

25 30 35 40

2010 24-15 28-22 32-32
-1-398 -2-473 -3-524 -45-410

Since water is compressible, though only slightly, the density depends on the pressure.
In the deeper parts of the ocean the pressures are enormous and have a considerable
effect on the density of the water. The change in unit volume of a material per pressure
unit is termed its compressibility coefficient /x. If the pressure unit is taken as 1 bar
(= 10*' dynes/cm^) then the compressibility coefficient of sea-water is of the order of
magnitude of 450 x 10"'; it increases somewhat with increasing pressure, increasing
salinity and increasing temperature and its extreme values lie somewhere between the
limits 510 and 390 X 10~". Ekman (1908) derived a precise empirical formula for the
effect of pressure on the density that takes into consideration the changes in the
compressibility coefficient with salinity and temperature (see Landholt-Bornstein,
1952; Dietrich, p. 484). This gives the density of sea-water for a given salinity, given
temperature and a fixed pressure and thus gives the density in situ a,^ ^ of a water
sample directly from a,.

Bjerknes and Sandstrom (1910) have presented complete tables to allow the spe-
cific volume anomaly or the density to be quickly found from the basic values for a
homogeneous sea at 0°C and with 35%o S for depths down to 10,000 m or pressures
of 10,000 decibars. Hesselberg and Sverdrup (1915) have given a method by which
the vertical variations in density can be calculated in a fairly simple way from the
temperature and the salinity. This simplification is due largely to the elimination of
part of the work by starting in the first place from the value for a,. If only the anomaly
is required, the tables prepared by Sverdrup (1933), which are still further simplified
and which give more accurate results, can be used. In general the relation a^ j^ =
^35, 0. 0 + S can be used where S is the specific volume anomaly. 5 is the sum of three
terms: 5 = A,j + Sgj, + S,,p. As shown by the indices the first term depends on the
temperature and the salinity, the others depend on the pressure and on one of the
other two factors each.

Since

The Sea-water and its Physical and Chemical Properties 43

S, X 10-3

and
then

"35. 0. 0 = 0-97264,

A^^^ = 0-02736

^.,t= 1 +

X 10-

1 + a, X 10-3

The values of the three terms J j,,, 6,,,^ and S,,^, can be given in short tables from which
the anomaly can be found correct to five decimal places. The same accuracy can be
obtained by accurate graphical methods or with the ingenious slide rule of Sund (1929).

The usual method for determining the density in oceanography is by calculation
from the temperature, the salinity and the pressure. The physical methods of de-
termining density such as the hydrostatic weighing and the pycnometer are unsuited
for oceanographic purposes, but the hydrometer has however often been utilized in
oceanography. Some very troublesome sources of error present with the ordinary
stem hydrometer have been discussed in detail by Krummel (1900), Buchanan (1884)
and Nansen (1900). They originate from insufficient attention to temperature differ-
ences between the instrument and the water sample and within the water sample
itself, the variable wetting of the instrument (traces of oil on the surface), the air
content of the water sample and not least to the variable capillary rise of the water
in the stem of the instrument which is often difficult to allow for. With proper use this
instrument gives values for a^ correct to two units in the second decimal place. Nan-
sen (1900) avoided the errors due to varying surface tension at the stem by using a
"hydrometer of total immersion" in which the ffoat is balanced in the water sample by
the addition of suitable weights. This method gives a^ correct to the third decimal
place (SvERDRUP, 1929). Since work with small weights is inconvenient on board
ship O. and H. Pettersson (1929), used a diff"erent method of loading a float hydro-
meter which is very simple and requires no handling of the float. A fine chain is sus-
pended from the float (chain hydrometer) so that the length of chain supported above
the bottom is a measure of the density.

Another method for the direct determination of the density which has been used in
older investigations (Pulfrich refractometer) utilizes the difference in refractive index
of the water sample from that of distilled water. This is measured either by the Hall-
wach method or by interferometry. The first method was used by Krummel (1889) on
the "Plankton" Expedition and later in 1892 by Drygalski on the Greenland Expedi-
tion. The interference method is more sensitive, although it requires suitable labora-
tory work to give the desired accuracy. (Askania Interferometer, Bein, Hirsekorn and
Moller, 1933, 1935). This interference method has been developed to give greater
precision and will give the density to the third decimal place in a^.

As well as the optical refractivity it is also possible to use the electrical conductivity
for the determination of densities. This method has several times been recommended
but has seldom actually been used. A survey of these experiments has been given by
Bein (1936). An instrument suitable for routine use was first developed by the Bureau
of Standards in Washington (Thuras, 1918; Wenner, 1930). It was in continual use
by vessels of the Ice Patrol in the North Atlantic Ocean from 1921 and was used by the

The Sea-water and its Physical and Chemical Properties

oceanographic vessel "Carnegie". Experience with this "saline tester" was not very
encouraging and the accuracy attained was, in spite of the greatest precautions, not
entirely satisfactory.

5. Vapour Pressure, Freezing Point, Boiling Point and Osmotic Pressure of Sea -water

Sea-water is a "dilute" solution and has the properties of such a solution. Due to the
low concentration of the dissolved material these will in several respects approach
those of the pure solvent, i.e. of pure water. It was shown quite early that the vapour
pressure p of a dilute solution is always less than the vapour pressure p^ of the pure
solvent and that the elevation of the boiling point is accompanied by a depression of
freezing point. As shown by Raoult and van't Hoff the relative lowering in vapour
pressure is independent of the nature of the material in solution and of the temperature
of the solution, and is proportional to the amount of dissolved material in solution in
the solvent. For a solution of « moles of a substance in Nq moles of a solvent:

Figure 22 shows the different phase states for pure water and for sea-water; it illus-
trates more clearly the relationship between the three well-known properties of dilute
solutions mentioned above. The curve G'S' showing the lowering of vapour pressure is
always lower than the vapour-pressure curve for pure water by the amount of the

760mm

r^/

/r /

\ '

Water phaseyp/

y i

/ z''

Ice

/ y

phase

y/ /

Sub cooled W ^

^y^ ^y^p Sea water j

u :r^^

yo-p^^

^L^'"

^-^

Vapour phase

/*(?■ tc Temperature fs h'
0°C IOO°C

Fig. 22. Phase states for pure water and for sea-water (schematic).

depression p^ — /;. Since for a given concentration (po — pMPq is constant, this de-
pression increases with increasing pressure and therefore also with increasing tem-
perature. The Po-curve for pure water cuts the line for a pressure of 760 mm Hg at
the point 5; this is the boiling point of pure water for which the corresponding tem-
perature ts = 100°C. The vapour-pressure curve for sea-water cuts this isobar first
at S' and this boiling point corresponds to a temperature z^- which is higher than /,.
The elevation of boiling point /1/s of sea-water of a given concentration is given by

At, = /v - r,.

77?^ Sea-water and its Physical and Chemical Properties

The depression of freezing point by a dissolved substance can also be inferred from
this diagram. The intersection G of the solid and liquid phases (the triple point)
corresponds to a temperature of 0-0075 °C. At 760 mm Hg the freezing point to of
pure water is 0°C and is fixed by the position of the intersection of the melting-point
curve Grwith the 760 mm isobar. It is the temperature at which the two phases (water
and ice) have the same vapour pressure, and therefore are in equilibrium with each
other. On the other hand, the freezing point of sea-water is at the intersection G' of
the vapour pressure curve for sea-water and that for ice; at this point the vapour
pressures over sea-water and over ice are the same. This corresponds at 760 mm Hg
to the freezing point of sea-water to' which is lower than to- The freezing-point de-
pression for sea-water is given by J/c = to' — to-

From this diagram it can immediately be deduced that both quantities Ate and At^
are larger the larger the value oi p^— p of the relative lowering of vapour pressure
ApJ p, that is the larger the concentration of the solution of the salinity. Quanti-
tatively it has been shown experimentally and theoretically that for low concentrations
the elevation of the boiling point and the depression of the boiling point are both
proportional to the concentration. In dilute solutions of substances termed in physical
chemistry "strong electrolytes", amongst which sea-water is included, it is found that
the electrolytic dissociation of the molecules is equivalent to an apparently larger
molecular concentration so that the simple proportionality no longer holds. The
accurate determination of saturated vapour pressures and of boiling points is experi-
mentally difficult and has been described in detail. The freezing point has been de-
termined by Hansen on eleven samples of sea-water, and by Knudsen (1903), using
determination of the constants, and the following empirical equation has been found

-0-0086 - 0-064633 a^ - 0000 1055 al.

This gives freezing temperatures correct to ±0003°.

Table 12. Freezing point and osmotic pressure of sea-water

Salinity (%)

Freezing point (°C)

-0-267

-0-534

-0-802

-1-074

-1-349

-1-627

-1-910

-2-196

Density (ctq)

3-96

8-00

12-02

16-07

20-10

24-14

28-21

32-27

Osmotic pressure

(atmos.)

3-23

6-44

9-69

12-98

16-32

19-67

23-12

26-59

Table 12 shows related values of salinities, freezing point t„ the density of sea- water
at this temperature and also the osmotic pressure (see later, p. 48). For the relative
lowering of vapour pressure Witting (1908) has given the equation

Aplp^ = 0-538 X 10-3 5.

the elevation of boiling point can as a first approximation be obtained from

At, = 0-01585.

Table 13 gives related values for the elevation of boiling point and the lowering of
vapour pressure at boiling point (at 760 mm Hg).

46 The Sea-water and its Physical and Chemical Properties

Table 13. Elevation of boiling point and lowering of vapour pressure in sea-water

Salinity, %„

J/,(X)

005

016

0-23

0-31

0-39

0-47

0-56

0-64

Jp in mm Hg at

760 mm Hg

213

4-23

6-45

8-47

10-73

12-97

15-23

17-55

A comparison of the temperature of the freezing point / a, that of maximum density
d and their corresponding densities at different sahnities is of some interest. Figure 23
shows the change in these temperatures with increasing sahnity. The temperature of

- \

\i>

24 695%<.J

h \.

-2

-4

-1 332-0 ->-

br^

20
S, %„

Fig. 23. Dependence of freezing temperature to on the salinity S.

maximum density decreases with increasing salinity more rapidly than the temperature
of the freezing point. At a salinity of 24-695%o (ctq = 19-839) both temperatures are
the same and

{) = fa = -1-332°C and a^ = a,^ = 19-852.

Reference might be made here to an oceanographic use of this (Helland-Hansen,
1911-12). Suppose a surface layer of a sea area is homo-haline with a salinity less than
24-695%o and that its surface is subject to strong cooling in winter. This cooling will
increase the density of the surface water, and as a consequence a vertical convection
must occur and will continue until the whole homo-haline surface layer reaches the
temperature of maximum density. It will then cease. The surface only will now be
cooled further by radiation until it reaches the freezing point and ice begins to form.
This will increase the salinity, and convection will again be set up and will be maintained
by the double effect of the increase in salinity and the decrease of temperature. These
conditions may occur, for example, in the Baltic. The homo-haline surface layer with
5 = 10%o during the winter cools and the vertical convection continues until the

The Sea-water and its Physical and Chemical Properties

temperature reaches +1-86°. The whole layer then has the maximum density a^ =
8-18. If cooling proceeds further the temperature falls only at the surface until this
reaches the freezing point tc — —0-53°, where ct,^ = 8-00, while the remainder of the
water mass remains at 4-1-86° and a^^ = 8-18. On further loss of heat ice is formed and
the density is raised by the liberation of salt until it reaches 8-18 when convection starts
again and continues as long as ice continues to form.

If, on the other hand, the surface layer has a salinity greater than 24-695% then the
vertical convection continues until the whole layer reaches the temperature of the
freezing point and proceeds further without interruption as long as fresh ice continues
to form. The difference between the two densities cr^g, and ua^ is, however, not large.
As shown in Fig. 24 these differences are largest at salinities of 6-7%o and very small
between 20%o and 35%o.

C-25

0-20

^ 015

's' 010
005

''"^

<>=

10 15 20 25 30 35 40
5, %o

Fig. 24. Density at the freezing temperature and maximum density of sea water as a function

of salinity.

Two adjacent water masses of different salinity will not, as far as their salinity is
concerned, be in equilibrium. In solutions of different concentrations in contact in
this way the material dissolved in the water will move from the region of higher con-
centration to that of lower concentration, that is, in the direction of the concentration
gradient. Known as molecular diffusion, it follows the same laws as thermal conduct-
ivity. If the salinity gradient is —{dSjdx) x 10~^, where S is given in %o, there will, by
diffusion, pass in unit time (sec) through unit area at right angles to the direction
of the gradient (1 cm-) an amount of salt Mg given by Mg = —K(dSjdx) x 10"^
where k is the molecular diffusion coefficient with the dimensions (g cm"^ sec~^).
The change with time in a given distribution of salinity follows from the differential
equation dSjct =^ k{c'^SIcx^) where /c is a constant independent of the time and the
distance (Fickian-diffusion equation). The diffusion coefficient for sea-water is very
small (0-0189 g cm"^ sec"^ at 35%o), molecular diffusion thus proceeds extremely
slowly, and long periods are needed to eliminate larger differences in salinity by pure
molecular diffusion. In this respect diffusion is quite analogous to thermal con-
ductivity.

Osmotic pressure is a phenomenon that is closely related to the properties of dilute
solutions described above. It is of very considerable importance for the biology of
living organisms in the sea. If a tank II (see Fig, 25) filled with sea-water of salinity

77?^ Sea-water and its Physical and Chemical Properties

S%o is separated from a tank I containing distilled water by a semi-permeable mem-
brane M which is permeable only for water and not for the substances in solution,
water will pass from tank I through the membrane M into tank II which contains the
the salt solution, and as a result the pressure in the tank II will rise. The sea-water could
be said to draw the pure water through the membrane. This process will continue until
the excess pressure in TI exceeds that in I by a fixed value P. This excess pressure at
which the system is in equilibrium is termed the osmotic pressure. According to physi-
cal chemistry it has been shown (see Nernst, Theoretische Chemie, 4th ed. 1903,

Fig. 25. For explanation of the osmotic pressure.

p. 157) that there is a relationship between the osmotic pressure and the depression
of freezing point which for sea- water at 0° takes the form P = —M-AAta- Stenius
(1904; see also Thompson, 1932) found the proportionality value 12-08 atm
for the constant in this equation. For other temperatures Pq must be multiplied by
(1 + 0-003670- Table 12 gives values for the osmotic pressure at 0° according to
Stenius.

The size of the osmotic pressure gives an idea of its biological importance. Or-
ganisms that live in the water are usually covered by a skin that is partly permeable
to water. They live in osmotic equilibrium with their environment. If one of these
organisms is placed in water of lesser salinity, water will pass in through its skin into
its body ; if the salinity is higher, water will be removed. Both processes, if they occur to
any extent, are unfavourable to the life of the organism since thecapacity of adaptation
is fixed within narrow limits.

6. Other Physical Properties of Sea-water

Other properties of sea-water that are also of importance in oceanography and
should be briefly mentioned are the heat capacity and the thermal conductivity, the
surface tension and the internal viscosity.

{a) The heat capacity of the specific heat of a body is the number of calories required
to heat 1 g of the material through 1 °C. The specific heat of pure water is dependent
on the temperature and shows a minimum of 0-947 at 34°C. It rises more rapidly to-
wards lower than towards higher temperatures and at 18°C it is 0-999.

A series of experimental determinations of the effect of the salinity was made by
Thoulet and Chevallier (1899) and their results have been utilized by Kriimmel to
prepare the figures shown in Table 14. The experimental value for the specific heat
of sea-water c^ is less than would be expected from the amount of salt in solution.

The Sea-water and its Physical and Chemical Properties
Table 14. The specific heat of sea-water at 17-5^

Salinity (°bo)
c„

0
1000

5
0-982

10
0-968

15
0-958

20
0-951

25
0-945

30
0-939

35
0-932

40
0-926

The dependence of Cp for sea-water on the temperature has not yet been closely
investigated, but presumably it is of the same form as that for pure water. Figure 26
shows the effect of temperature on Cj, for pure water and for sea-water with 35%o S.
The dependence of c^ on the pressure/? can be found using well known thermodynamic

I-OI
100

^ure

wate

:a w

Iter

3-99

0-95

0^4

0-93

10 20 30 40 50

r, "C
Fig. 26. Specific heat for pure water and for sea water at 35o/(,p salinity.

principles (Ekman, 1914). If the pressure/? is taken in decibars, and the density of the
water is p, the absolute temperature T, the coefficient of thermal expansion /S, and J
is the mechanical equivalent of heat (4-1863 x 10^ ergs/cal or dyn cm/cal), then

dp pj \ 8t

Ekman has calculated the value of c^ for atmospheric pressure and for pressures from
p = 2000 top = 10,000 decibars, corresponding to depths of about 2000 to 10,000 m
(Table 15). At great depths c^ differs appreciably from 1 and this must be taken into
account in accurate theoretical calculations.

Table 15. Specific heat of sea-water at different pressures when ct = 28 (34-8%o)

Temperature

' (^C)

-2

Pressure in

dbar

0-942

0-941

0-938

0-935

0-933

0-932

1000

0-933

0-930

0-929

0-928

0-927

2000

0-925

0-924

0-923

0-922

0-921

3000

0-910

0-912

0-913

—

6000

0-898

0-901

0-904

—

8000

—

0-892

0-896

—

The relationship k = c^jc^ is also of interest. The specific heat/constant volume
c,; is a little less than Cp. From thermodynamics the equation

Cp = c„ + ^^

pixj

50 The Sea-water and its Physical and Chemical Properties

can be derived, where jj. is the cubic compressibiHty. For sea-water where Oq = 28
(34-84%o S) at temperatures of 0" and 30°C respectively, /3 = 15 x 10^« and 334 x
10-« grad-i and ix = 46-59 x IQ-^^ and 42-07 x lO-^^ jyn-i cm^. From this it can be
found that k -= 1-0004 and 1-0207 for 0°C and 30°C respectively. At greater depths ^J.
is smaller and there k is larger than at the surface.

(h) The thermal conductivity coefficient A is defined by the equation

Q = -x(ddidx),

where Q (cal/sec) is the amount of heat passing through 1 cm- at right angles to the
flow and dd ( C) is the change in temperature along a distance d.x (cm) in the direction
of flow. A thus has the dimensions (cal cm~^ sec~^ grad"^). For pure water A =
0-001325 + 4 X 10-«/.

A has not been determined directly for sea-water; as a first approximation, according
to Weber's rule, the ratio of the thermal conductivities of two substances is the same
as that of the thermal capacities of equal volumes. This gives the values shown in
Table 16 for the coefficient of thermal conductivity for different salinities.

Table 16. Coefficient of thermal conductivity at different salinities

Salinity (%„)

10 I 20

30 I 35 40

Thermal conductivity I !

coefficient (X 10-») 1-400 1-367 1-353 • 1-346 ! 1-341 1-337

For oceanic water (35%o S) the thermal conductivity coefficient is about 4-2% less
than for pure water. The temperature conductivity coefficient is the quantity a = XJipCp)
and has the dimensions (cm- sec~^). For sea-water pCj, is not very different from 1 and
the numerical difference between A and a is slight.

(c) In fluids with motion there is a shear stress between every layer in the direction
of flow and the adjacent parallel layer, and this shearing stress is proportional to the
velocity gradient perpendicular to the direction of flow, that is

dv
^ dz

The proportionality factor /x is a measure of viscosity or inner (molecular) friction
(g cm~^ sec~^). For many flow phenomena there occurs the coefficient i- = /x/p,
the kinematic viscosity (cm- sec "^). These frictional coefficients decrease rapidly
with increasing temperature. For pure water, the values shown in Table 17 are ob-
tained. According to the investigations of Krummel and Ruppin (1905) viscosity
increases very little with salinity; at 0°C by 3-9 or 5-2% for 25%o S and 35%o S re-
spectively and at 30X' by 6- 1 or 8-2"o. The effect of pressure appears to be negligible.

Table 17. Viscosity coefficients for pure water
(g cm"i sec"^)

Temperature ( C) 0 10 j 20 30 40

/x 0-0179 0-0131 , 0-0100 1 00080 00065

The Sea-water and its Physical and Chemical Properties 51

The magnitude of the molecular viscosity was eariier attributed some importance
in the biological and dynamic processes in the sea, but it has since been recognized
that processes in oceanic currents are always turbulent and the coefficient of turbulent
viscosity is considerably larger than the coefficient of molecular viscosity. This has
very much reduced the importance of the latter.

(d) Surface tension. Krummel (1907) investigated the dependence of the surface
tension on the temperature and the salinity; it decreases with rising temperature and
with decreasing salinity. Fleming and Revelle (1939) have taken more recent values
to derive the equation

surface tension in dyn/cm^ = 75-64 - 0-144/ + 0-0399 CI.

Impurities in the water always lead to a considerable reduction and this must be taken
into consideration for surface waters of the sea.

7. The Optical Properties of Sea-water

(a) The Extinction of Incoming Radiation
Parallel radiation entering a layer of sea-water is gradually weakened in three ways:

(1) By absorption by the pure sea- water.

(2) By scattering by the pure sea-water.

(3) By scattering, diffraction and reflection by suspended particles in the water
(impurity of sea- water).

The last two factors do not change the form of the energy but divert a part of the
radiation from its original direction. A beam of radiation of wavelength A passing
through a distance dx in water is reduced in intensity by an amount dl which is pro-
portional to the intensity and to the distance ^.v travelled through the water, so that
dl = —Kidx. K the extinction coefficient (cm~^) is dependent on the wavelength A.
If the intensity of the radiation is /q when x = 0, then for a distance ,v

I = I,e-^\

The reduction in intensity of the radiation is often characterized in practice by the
extinction E for a layer of thickness 1 m and is given as a percentage of the incident
radiation

£■= 100 ("l - ^

The transmission D may also be used, and gives the percentage of the incident radia-
tion passing through a layer of fixed thickness

i) = 100 - - 100 e--^^

Detailed measurements have been made of the extinction coefficient for water over
the whole spectral region from 0-186/x in the ultraviolet to 8-5 ju. in the infra-red.
The spread of 2-3% in the values obtained in different series of measurements are
largely due to the difficulty of preparing "pure water". Dietrich (1939) has given a
comparison of the older measurements of Aschkinas (1895) and more recent values
by Kreusler (1901), Sawyer (1931) and Collins (1925, 1933) from which the values
shown in Table 1 8 have been abstracted.

The Sea-water and its Physical and Chemical Properties
Table 18. Absorption coefficient k (cm"^) for pure sea-water

Wavelength

X in fj. \ K

Ain/i

A in /x K

Ain/x

0-20

000899

0-70 00084

1-30 1-50

200

0-30

000151

0-80 00240

1-40 i 160

2-10

0-31

00084

0-90 f 00655

1-50 i 19-4

2-30

0-40

000072

100 0-397

1-60

8-0

2-40

0-50

000016

110 0-203

1-70

7-3

2-50

0-60

000125

1-20 1-232

1-90

2-60

100

0-2-0-3 /x according to Kreusler (1901), 0-31 -0-60 /x according to Sawyer (1931) and from 0-7 /n
according to Collins (1933).

Figure 27 shows the spectral range from 0- 1 86 /x to 2.65 /x. From about 0-48 fi towards
the red end of the spectrum and beyond, the absorption coefficient increases strongly
and continuously. According to the measurements of Aschkinas, weaker absorption
bands follow stronger bands between 2-86 /x and 3-27 /x, and at 6-7 ij. where there is
almost complete extinction of the radiation. The absorption depends slightly on the

0-01

100-0

60-0
400

J .-^

\ /

200

■

K /

10-0
e-0

\ /

; visit

le spectrum

1-0

0-6

0-4

/**-J

;

0-2

0-1

006
004

;

002

•

0-01

0006
0 004

0 002

0-001

0-0006
00004

00002
-\.n.r,r,\

0-5 1-0 I-!

\ in /i

2-0

2-5

100

1000

10000

Fig. 27. Absorption coeflRcient for pure water (pure sea water for parallel radiation (wave-
length range 0-186-2-65 /j.) (From 0-2 to 0-3 ^ according to Kreusler; from 0-31 to 0-60 /u.
according to Sawyer; from 0-70 /x on . . . according to Collins).

temperature and an effect of the salinity has been found but from the summary given
by Dietrich it can be seen that the absorption in pure sea-water is almost the same as in
pure fresh water.

The extinction coefficient k takes account of the effects of both scattering and ab-
sorption. The scattering of light in a turbid medium is caused by reflection and diffrac-
tion of the incident light by the small particles suspended in the medium. If the size

The Sea-water and its Physical and Chemical Properties

of these particles is very small compared with the wavelength of light and if the con-
centration is not too large, the scattering is due to pure diffraction following Rayleigh's
law; according to this the reduction in intensity of the incident light is inversely pro-
portional to the fourth power of the wavelength. Amongst the phenomena due to
scattering is included that known as the Tyndall ejfect, where a beam of light passing
through a turbid medium produces a more or less intensive illumination of those por-
tions in the medium affected by light. This is due to reflection and scattering of the
light by the suspended particles. Since the shorter wavelengths are more strongly
scattered, the Tyndall-light is bluish. The water molecules themselves can be regarded
as scattering particles. Thereby one thought to explain also the blue colour of the
scattered light in pure water. However, it has later been recognized that a direct scatter-
ing by the water molecules can hardly occur since there are too many compressed
into a small space and the distances between them are too small relative to their
diameter. According to the theory of Smoluchowski irregular molecular movements
give rise to an optical inhomogeneity (streaks; Schlieren) of very small dimensions
and are therefore responsible for the scattering of light.

Table 19. The energy distribution in the spectrum of sunlight after passing
through water layers of different thickness

Wave-

Thickness of the water layer

length
0^)

001

100

0-2-0-6

237

236

229

172

0-6-0-9

360

359

353

305

129

0-9-1 -2

179

178 1 172

123

1-2-1 -5

82 i 63

1-5-1-8

1-8-2-1

—

2-1-2-4

—

2-4-2-7

—

2-7-3-0

0-4

0-2

—

Total

10000

993-7

952-1

859-4

730-2

549-3

358-1

181-5

13-9

The only natural parallel radiation occurring in the upper surface of the sea is
direct sunlight. On passing through water the spectrum of sunlight undergoes great
changes. Schmidt (1908), on the basis of the extinction values of Aschkinas and values
according to Langley for the distribution of radiation energy from the sun on the
surface of the sea, has calculated the spectrum of the sunlight at different depths and
obtained the values given in Table 19 for water layers of difiTerent thickness; the total
radiation from the sun incident on the surface of the sea is taken as 1000 (Fig. 28).
The total extinction for different layer-thickness is given in Table 20. The reduction
in intensity of sunlight after passing through very thin layers of water is quite consider-
able. For a layer 1 cm thick, wavelengths >l-5^t are completely eliminated and the
spectrum extends only to 0-9 /x. For layers 100 m thick the remaining energy has
fallen to less than 1-5%.

The Sea-water and its Physical and Chemical Properties

Wove length, //

Fig. 28. Energy distribution in solar radiation after passing through water layers of different
thickness (according to Schmidt). A-B, at the water surface; A-C, after passing through
1 cm of water; A-D, after passing through 1 m of water; A-E, after passing through

100 m of water.

Table 20. Extinction values for sunlight passing through sea-water

Down to a depth of j 00 1 mm
Extinction in per cent' 0-6

01 mm ] 1 mm
4-8 i 14-1

1 cm
270

10 cm

45-1

1 m

64-1

10m

81-8

100 m

98-6

The extinction coefficients in Table 1 8 are valid only for pure sea-water. The water
of the sea is, however, not optically pure, and always contains more or less large
amounts of suspended organic and inorganic particles. The intensity of the light
passing through the water is still further reduced by scattering on these particles as
well as by the ordinary extinction. It may be so strong that the actual absorption,
especially in the presence of very small particles Rayleigh's law applies, but for larger
particles the scattering is almost independent on the wavelength. It depends primarily
on that part of the total surface influenced by the sun radiation of all the individual
particles present in a unit volume. Scattering by large particles is then no longer
colour selective (Pernter, 1901).

The reduction in the intensity of radiation in the sea under natural conditions has,
for the first time, recently been subjected to more accurate investigation, because of
its special biological interest (see especially Jerlov, 1951; Joseph, 1952). These
measurements have been made principally with photo-electric cells which have a
sensitivity extending over a considerable range of wavelengths, while the extinction
coefficients mentioned above were measured by spectrobolometric methods. The re-
sults are thus only comparable after appropriate corrections. The most detailed
measurements have been made on lakes (Sauberer and Ruttner, 1 941) ; measurements
in the sea which are of greater interest in the present connection are rather few in
number. The extinction coefficient applies to the solar radiation and the diff'use sky
radiation taken together. When radiation passes through water it undergoes a pro-
gressive alteration both qualitatively and quantitatively. The long wave and short
wave parts of the spectrum are filtered out almost at once so that the light soon takes
on a bluish-green or blue colour. With a greater degree of optical impurity the effect
of the scattering is less colour selective; the remaining light is more greenish, or with
strong turbidity even yellowish green (Pettersson, 1936). At the same time the light

The Sea-water and its Physical and Chemical Properties

undergoes a progressive change in direction since the most obUque light is diminished
most while the diffuse light formed by scattering increases continuously.

The first light measurements on the open sea were made by Poole and Aitkins
(1924). Detailed measurements have been made more recently by Clarke (1933, 1936,
1938) and by Clarke and Oster (1935); (see also Utterback 1936). For an example
Figs. 29 and 30 show the percentage reduction in intensity of light in different parts
of the spectrum for the surface layers of the Sargasso Sea and of the Gulf of Maine.

Percentage of surface light
OOI 005 0-1 05 1-0 50 10 50 100

20
40
60
80
^ iOO
120
140
160
180

1 1 1

mi:

1 1 1

iin

" 309,

Red

}

Green ji

■y

'^Violet i

Bju^

■^

3i2ldr

"^310

Fig. 29. Decrease in the intensity of light in the Sargasso Sea for different spectral ranges as
a percentage of the intensity at the surface (according to Oster and Clarke).

OOI

Percentoqe of surface light
005 0-1 05 10 5 10

50 100

_ 1 1 1

1 III

- 1 1 I 1 111

' "!""

\ '

.-^'"''xfd

Red j^

E ^

J20

BlueV''^

Bor^iolet

319^' 32lK

-een

Fig. 30. Decrease in the intensity of light in the Gulf of Maine for different spectral ranges,
as a percentage of the surface intensity (according to Oster and Clarke).

As a striking feature the extinction curve is almost linear with depth so that within
the spectral region investigated the extinction coefficient is almost constant and is
independent of the depth. The violet and the blue are most strongly affected by the
turbidity, the red is least affected.

The extinction coefficient for shelf and coastal water is considerably larger than for
ocean water, approximately two to three times larger or even more. Its size represents
only the order of magnitude of the coefficient since these types of water show large
variations both in time and locality. Swedish light measurements, which have been

56 The Sea-water and its Physical and Chemical Properties

made, principally by the Oceanographic Institute in Goteborg (Pettersson), since
1933 in fiords, in the Skagerrak, in the Kattegat, and in the Baltic have given similar
results, but they also show a particularly strong dependence of the reduction in in-
tensity of the light near to the thermocline (discontinuity in vertical density distribu-
tion). This intensification of the extinction is undoubtedly due to an enrichment of
suspended particles at such layers. This enrichment shows considerable local diff'erences
and causes strong variations in the extinction coefficient. If the scattering and the
absorption due to the suspended particles is removed by filtering the water samples
there remains a selective absorption which must be due to strongly absorbing humic
material dissolved in the water. This "yellow material" must be an organic metabolic
product, either from the land or from the remains of decomposed plankton. The
turbidity of the water can now be determined continuously from a moving ship by the
self-recording transparency meter (Joseph, 1950, 1952) and the results can be used in
suitable cases to determine the origin of a water mass since the extinction value pro-
vides a persistent characteristic (Dietrich, 1953; Joseph, 1953; Jerlov, 1953; see
also Wyrtki, 1950). The distribution of particles in suspension can be studied with
the Tyndall-meter which measures the intensity of the scattered light produced from
a parallel beam of light, by comparison with the known intensity of an illuminated
glass filter using a Pulfrich photo-meter. This apparatus can also be used for the
measurement of the scattering from suspended and dissolved material in especially
transparent ocean water, corresponding measurements of this type have been made
by Jerlov (1953) in the three oceans during the "Albatross" Expedition.

{h) Refraction and Reflection of Radiation

Parallel radiation incident on the surface of the water will be partly reflected and in
part will enter the water. The angle of reflection will be the same as the angle of inci-
dence but the ratio of the intensities of the incident and the reflected beam will be
dependent on the angle of incidence of the original radiation itself. Radiation entering
the reflecting medium undergoes a change of direction on passing through the surface,
and the angle of this refracted beam is given by the equation

sin /

-^ — = n,

sm r

where / is the angle of ncidence, r is the angle of refraction and n is known as the
refractive index. For air and pure water it is almost exactly 1-333338 or -^4/3. That is,
in water which is optically denser the beam is refracted towards the perpendicular
(Fig. 31). The refractive index for a ray passing from the water into air is Xjn ~ 0-75.
If the angle of incidence of radiation passing from the water into air increases, the
angle / will increase faster than the angle r until finally the value of / reaches 90°;
the outgoing ray then passes along the surface of the water. This occurs when r =
48-5° = R (see Fig. 31). If/- increases still further, radiation cannot enter the air but is
reflected entirely within the water; R is known as the critical angle for total reflection.
SoRET and Sarasin (1889) have measured the refractive index of mediterranean
water (approx. 37%o S) for various wavelengths and compared these values with those
for pure water. Table 21 shows the results. The dependence on salinity is, however,
suflUciently large for use in the optical determination of salinity (refractometer) ;

The Sea-water and its Physical and Chemical Properties

wm'////myMMM'/M//m}/m'M-

Fig. 31. Reflection and refraction of radiation at the interface between air and water.

Table 21. Values of the refractive index for sea-water and for pure water
(After SoRET and Sarasin, 1889)

Frauenhofer

A
in /i

PureH^O
/ = 20°C ■

Sea-water 37%o

Sea -water-

line

20 C

10 C

20 ^C

c 1

^ i
F 1
h

0-6563
0-5896
0-4861
0-4102

1-33120
1-33305
1-33718
1 -34234

1-33816
1-34011

1-34437
1-34973

1-33906
1-33092
1-34518
1-35064

000696
000706
000719
0-00739

besides this it is much stronger dependent on the temperature. More recent investiga-
tions on the dependence of the refractive index of sea-water on the temperature and
saHnity have been carried out by Bein (1935) at the Physikalisch Technische Reichs-
anstalt in Berlin. Table 22 shows the deviation of the refractive index of sea- water «s
from the refractive index for pure water, /z„. = 1-333338 (at 15°C, A = 587, 6 m/ii) at
different temperatures and sahnities. The dependence on the wavelengths of the light
used is not as large and only has to be taken into consideration for more accurate
treatment.

Table 22. Variation of the refractive index {n^ — «„.) x 10®

with temperature and salinity

(According to Bein, 1935)

. /^C

5%„\

4001

3814

3697

3621

3594

3571

4989

4759

4617

4524

4491

4463

5977

5708

5538

5429

5390

5357

6966

6657

6463

6337

6292

6254

7956

7610

7391

7250

7199

7157

The Sea-water and its Physical ami Chemical Properties

The relationship between the intensities of the incident and the reflected radiation is
expressed by Fresnel's law. If 7 is the intensity of the incident radiation and R that
of the reflected radiation, the relationship between them is given by

R
J

sin^ (/ — /■) tg^ (i

sin2 (/ + /-) tg^ (/ + /-)

Ify = 100 and n == 1-333 this gives the values shown in Table 23. If the angle of inci-
dence is 0°, only 2% of the radiation is reflected and almost the whole of the energy
penetrates through the surface.

Table 23. Reflected radiation R and refracted radiation Dfor different angles of incidence
i of radiation on a water surface {J = 100, n = 1-333)

0°

10°

20°

30°

40°

50°

60°

70°

80°

100^

0°

7° 29'

14° 52'

22° 02'

28° 50'

35° 05'

40° 31'

44° 49'

47° 38'

48° 35'

2-1

2-5

3-4

13-4

34-8

1000

980

97-9

96-6

940

86-6

65-2

With increasing angle of incidence the reflected energy increases only slowly up to
about / = 60^ and thereafter very rapidly. The larger the angle of incidence the more
is reflected, at 70° more than 13%, at 80° more than 35%. This is shown in Fig. 32.
The rays coming from the upper left incident on the surface are split into reflected and

10° f

Fig. 32. Graphical representation of the proportions of reflected and transmitted radiation

incident on the surface of water at different angles. For each ray incident from the upper

left with an intensity of 100 there will be a reflected ray and a ray transmitted into the water.

Both are represented by vectors which give the intensity and the direction of the ray.

entrant rays; the incident rays have an intensity of 100 and the vectors marked on the
diagram correspond in intensity and direction to the reflected and entrant rays. Larger
values for the reflected energy only occur with obliquely incident light and especially
in that range, where the entrant radiation falls to very low values. It can be seen that

The Sea-water and its Physical and Chemical Properties 59

the direct incident radiation coming from a whole quadrant is concentrated into a
fairly narrow beam range from 0° to 48-5°, while at angles of incidence more than 65^
the intensity of the entrant radiation is rather small. Schmidt (1908) showed by
actinometric measurements at the surface of pure water that the same conditions apply
for the total solar radiation as for the D line of sodium (n = 1-333). More recent
measurements by Poole and Atkins (1926) and by Whitney (1938), as well as by
Angstrom (1925) using the pyranometer, show that the theorectical values for re-
flection are also obtained essentially in practice. However, the reflection is more or
less strongly increased by waves on the surface of the water; it may be increased in this
way by more than 50% (Lauscher, 1944).
(c) The Behaviour of the Water Surface for Diffuse Incoming and Outgoing Radiation

As well as the direct sunlight, which may be regarded as unilateral parallel radiation,
there is also a general diffuse radiation for which conditions relative to the sea surface
are rather different. The diffuse radiation on the surface of sea includes: (1) diffuse sky
light (daylight) which is essentially short-wave radiation (between 0-38 ju and 0-75 /^i)
and is only present in the day time; and (2) the long-wave radiation from the atmosphere
which is long-wave (maxima at 7-5 /z and 12-5 /x), and is present both day and night.
Each single beam of the diffuse radiation that is incident on the surface of the water at
an angle / is partly reflected following Fresnel's law and is thus subject to a corre-
sponding reflection loss as shown by the values given in Table 23. Since the diffuse
radiation comes from all directions and the radiation with a greater angle of incidence
is more strongly reflected, it is necessary to find the sum of the losses for each angle of
incidence in order to determine the total loss by reflection. The calculation of this
total from the values r(i) given in Table 23 gives the reflection losses (forn = 1-333) as
0-660, that is 6-6% of the diffuse radiation is reflected from the surface of the water.
Considering the refractive index to be slightly different for different parts of the
spectrum this value varies between 5% and 10%.

Mention should also be made here of the properties of water as a source of radiation
(Schmidt, 1915). Since the extinction coeflftcient of water for long-wave radiation is
particularly large and the thermal radiation from the surface of the sea contains only
longer wavelengths (around lO^u) it can be expected from Kirchhoff's law that as a
source of radiation water would behave as a black body. Nevertheless, water radiates
less than a surface of the same temperature since each beam coming from the interior
of the water mass will suffer a reflection loss at the surface which will reduce the
intensity of the total from the surface outgoing radiation (Fig. 33).

In addition to this reflection loss the intensity of the radiation suffers a further de-
crease since in passing through the surface to the air it must spread out into a larger
space. The radiation from water within a space angle of 2 x 48° 35' = 97° 2' is spread
out over a full 180°. If this is taken into account (Schmidt, 1916) it is found that for
a temperature range of 0-20°C the outgoing radiation from a water surface is about
9-10% less than that from a black surface. Since the radiation from a black body
according to the Stefan-Boltzmann law is given by £" = aT'^ where a = 1-374 x 10~^-
cal cm"2 sec"^ grad"^ the radiation from a flat water surface will be given by ^4 =
0-904CTr''. Angstrom has found experimentally that for long-wave radiation the effici-
ency of emission of sea-water is 96% of that of a black body. The constant in the above
equation should therefore be not very different from 0-95 for the temperature range

The Sea-water and its Physical and Chemical Properties

Fig. 33. Back radiation from the interior of the sea towards the water surface.

concerned. Lauscher (1944) has obtained the same result in another way and found
the value 0-9535 for the constant. Falkenberg (1928) has made similar calculations
and has found the somewhat lower value 0-937 for this constant.

{d) The Colour of Sea-water

The colour of sea-water in the scientific sense is taken to include all those colour
phenomena which arise because of the optical properties of sea-water and the sub-
stances dissolved and suspended in it. The colour of the sea can vary widely and may
assume any shade from a yellowish green to the deepest blue. To observe the colour
of the sea undisturbed by external reflections it is best to look through a tube which is
blackened inside, dipped in the water. The colour can be determined by comparison
with standard colours or by spectrophotometry. Kalle (1938) has designed a special
colour measurement tube in which the colour of the sea can be determined with a
comparator. In practice, the colour is for preference determined with standard
colours, using the Forel-Ule scale. Accurate colour determinations in the open sea are
by no means frequent and have been made almost only by oceanographic expeditions.
The largest part of the surface of the ocean is blue {Forel 1 and 2), particularly, the
regions within the tropics and subtropics, while the green colour is prevalent in coastal
areas and shallow seas, especially in adjacent seas and polar regions. In the
Atlantic Ocean (Schott, 1942) there is a certain symmetry in the distribution of colour.
From 15° to 35° N. and from 10° to 30° S. it is a deep blue. The purest and richest
colour is in the central parts of these areas, roughly from the Bermudas to near
Madeira and off the Brazilian coast till St. Helena. In the Benguela current, generally
in areas of upwelling, for example off the West African coast in the north and off the
south-west African coast in the south the sea-water has a more greenish colour. In the
Southern Hemisphere a tongue of greenish blue water runs from this coast of South
Africa far up to the north between 0° and 10° S. (up to St. Paul Island).

The higher latitudes in both hemispheres are always discoloured. Greenish blue
predominates north of 40° N. and gradually changes to green. The waters of the English
Channel, the North Sea and the Baltic are of the same colour. In the Southern

The Sea-water and its Physical and Chemical Properties 61

Hemisphere the colder water of the Falkland current and the oceans areas around
Bouvet Island are mostly greenish blue to green.

An explanation of the colour of pure sea-water must be sought, in the first place,
in the optical properties of sea-water. The Bunsen theory ascribed the blue colour of
the sea to the combined effects of the spectral absorption of pure sea-water and re-
flection by the particles suspended in the water (absorption theory). The light entering
the water (direct sunlight and diffuse radiation from the sky) will be weakened least
in the blue by absorption. Down into the deeper layers the light becomes more and
more blue. This relative concentration of blue is further increased in the light reflected
from small particles and passing back to the surface, the light returning through the
surface is thus blue. Against this absorption theory, Soret has set a diffraction theory
according to which the explanation of the blue colour of the sea is analogous to that
of the blue colour of the sky and is due to the scattering of light in the water. Ramana-
THAN (1923) has attempted to prove by experiment and theoretical investigation that
pure sea-water should show an indigo blue colour by molecular dispersion and by
selective absorption, and that small amounts of suspended matter have little effect on
the colour. According to the theoretical investigations of Gans (1924), the colour is
due principally to diffraction of higher orders (see also Lauscher, 1947).

A third possible explanation for the widely occurring greenish colour was advanced
by WiTTSTEiN (1860) and later by Spring (1886, 1898) in the so-called "solution
theory". In this, blue was regarded as the actual colour of the water and all variations
were due to different substances dissolved in the water. This effect was ascribed prin-
cipally to organic humus materials that in increasing concentration made the water
first green, then yellowish green and finally, in extreme cases, brown.

It was first pointed out by Kalle (1938, 1939) that the physiology of colour vision
must play a large part in the explanation of the colour assumed by the sea and must be
taken into consideration. According to the Young-Helmholtz theory of colour vision,
the human eye has three groups of colour-sensitive elements (cones), each of which is
sensitive to one of the three primary colours, red, blue and green. The stimulation of
two or all three of these groups at the same time gives the impression of a mixed
colour. Every different colour impression is produced by a definite ratio in the strength
of the stimulation of the three different types of cones. A "colour triangle" (Fig. 34)
can be used to represent diagrammatically all possible colour impressions. The three
corners of the triangle represent the total (100%) stimulation of only one group of
receptors — red, green or blue. At every point on the triangle the sum of the oblique
co-ordinates of the point is always 100%, and these co-ordinates represent the per-
centage composition of the mixed colour characterized by that point. The point
W = white which, by definition, is composed of a mixture of 33J% of each of the three
primary colours hes at the centre of gravity of the triangle. All tones of the same colour
lie along a straight fine that runs radially from the white point; the nearer a point on
such a line lies to one of the sides of the triangle the more saturated is the colour it
represents. The position of the spectral colours within the triangle is shown by the
curve marked on the diagram. Since the spectral colours are the most saturated
colours possible in nature, all colours found in nature must lie on the area within the
spectral curve and the line joining its two end-points.

In the light of the consequences of this theory, Kalle has investigated the effects

The Sea-water and its Physical and Chemical Properties

of selective absorption and selective scattering and also of the interaction of these two
processes on the colour of the sea. These results are summarized in Fig. 35 which
shows a part of the colour triangle and the spectral curve. The absorption colour of
sea-water lies on a curve running from the white point and approaching concave

Fig. 34. Colour triangle of the Young-Helmholtz colour theory and spectral curve.

Fig. 35. Part of the colour triangle showing colour points for sea-water colour.

The Sea-water and its Physical and Chemical Properties 63

downwards the spectral curve asymptotically. With layers of increasing thickness
the increasing saturation of the colour gives a slow displacement towards the blue,
while at the same time the brightness of the colour decreases rapidly so that only a
relatively thin surface layer is concerned in the colour of the sea. According to Kalle,
the result is a colour with a wavelength approaching 492 m/x, a somewhat greenish
blue, corresponding to a light path of 38 m. This shows immediately that the deep
blue colour of the Sargasso Sea cannot be explained in this way. If selective scattering
of the different colours is taken into account the colour curve lies further towards
shorter wavelengths. As far as the colour is concerned the most important point on this
curve approaches that corresponding to a 50 m thick layer where the colour value
is 485 m/Li. This value agrees fairly well with the colour of the Sargasso Sea, especially
if the higher order scattering which would give a further slight displacement towards
shorter wavelengths is taken into account. The absorption and the scattering of light
are thus responsible for the blue colour of the tropical and subtropical areas of the
ocean and they are reinforced by the greater brightness of the sunlight and of the diffuse
light from the sky and by the almost completely pure sea-water of these areas.

For water masses that are not so pure and contain large numbers of suspended
particles (mostly plankton), as is usually the case in higher latitudes, the depth from
which the selective scattering is reflected is less, and the colour gradually reverts to a
value of 495 m/x. This would be more or less the longest wavelength for the colour
of the sea if only absorption and scattering were involved during its formation. Other
causes are, however, required to explain the greenish colours of longer wavelength
than 495 m/i that are also of frequent occurrence in the open ocean. Investigation has
shown that these are due to coloration caused by yellowish substances dissolved in the
water. These substances appear to be related to humus and are apparently to be re-
garded as products of phytoplankton metabolism. They displace the colour of the
water towards the green especially in water masses such as in the English Channel
and in the North Sea where values of 498 m/x to 505 m^u may occur. In coastal regions
further humus material carried by fresh water flowing into the sea from rivers is
added to the more oceanic yellow material and causes a further displacement towards
yellow-brown colours. In addition to these yellow substances there may also be
fluorescence phenomena in the seas as Ramanathan, and later Kalle, believed; these
would give a further displacement towards the green but the extent to which such fac-
tors are present is not yet certain.

A qualitative survey of the contribution of each single factor to the colour of the
sea has been given by Kalle in Fig. 36. In the clearest water and with a depth of visi-
bility of 50-60 m, selective scattering plays to a very large extent the principal part.
If cloudiness due to the presence of plankton occurs, the depth of visibility gradually
decreases and the natural absorptive colour of water which tends towards a greenish
shade begins to predominate. At the same time small amounts of yellowish substances
may be formed as the colour tends more and more towards green. With the increasing
turbidity the yellow material becomes more and more important until finally, at very
small depths of visibility, the discoloration is due to the natural colour of the material
causing the turbidity. Very close to the coast the natural colour of the bottom begins
to show through the shallow water, and the colour of the water is clearly altered to-
wards this.

The Sea-water and its Physical and Chemical Properties

Turbidity
discoloured

70ny/F60.
;-5l5m/^F25.

|i-488ny/F5.

477ny/ FO.

Magnitude of porticipotion
of the individuol factors
giving rise to ttie colour of
the sea

Fig. 36. Quantitative representation of the contribution of the individual factors giving rise
to the colour of the sea (according to Kalle).

8. The Chemistry of the Sea

In general, liquids have the property of absorbing gases with which they are in
contact to give a solution of the gas in the liquid. The solubility of the gas in the liquid
is not unlimited, but usually fairly soon reaches a limit; the liquid is then saturated.
According to Henry's law the amount of gas dissolved in a saturated solution is pro-
portional to the pressure of the gas in contact with the liquid. If the liquid is in contact
with a mixture of gases then each separate gas is absorbed according to its partial
pressure. When the liquid is completely at rest the process of solution depends on the
process of diffusion, and thus requires time for the pressure of the gas in the liquid
to come to the same pressure as the gas outside it. In nature, the wave motions,
turbulent currents and convections can accelerate considerably the uptake of gas by
the liquid. By the gas content of a sample of water is understood the amount of gas
in the water expressed in volume units (ml/litre) at NTP (0°C and standard pressure
of 760 mm Hg). The actual gas content may of course differ more or less from the
amount present when saturated.

Besides, by this absolute definition the gas content may also be characterized by the
ratio of actual content to that by saturation. It is then specified by the relative gas
content which is expressed in per cent of the amount required for saturation. The
absorption coefficient is taken as that volume of gas which can be absorbed by unit
volume of liquid at a given temperature and standard pressure.

The Sea-water ami its Physical and Chemical Properties

If the solution process is limited to purely physical absorption the absorbed gas
does not enter into chemical combination with the water; the situation is then fairly
simple. It is, however, possible for the gas to combine chemically with the liquid.
Both possibilities occur in the atmosphere-ocean system. Oxygen, nitrogen and the
rare gases obey the pure physical absorption ; carbon dioxide, on the other hand, fol-
lows the second possibility since it reacts both with the water itself and in part also
with the salts dissolved in it. (For chemistry of sea waters see especially Harvey, 1955.)

(a) Oxygen, Nitrogen and Hydrogen Sulphide Contents of Sea-water

The composition of the air absorbed by pure water can be calculated from the
absorption coefficients of the gases present in the atmosphere and is shown in Table 24
for 0° and 30°C. It is different from that of atmospheric air since the absorption coeffi-
cient of the individual gases is very different. In atmospheric air the ratio of oxygen
to nitrogen is 21 : 78 or about 1 : 4, but in the air dissolved in water at 0°C it is 35 : 62,
and at 30°C 33 : 64 or about 1 : 2. The air dissolved in water is thus twice as rich in
oxygen as atmospheric air, but it should not be forgotten that while a litre of air con-
tains 210 ml of oxygen, a litre of water saturated with air contains only about 10 ml.

Table 24. Distribution of atmospheric gases at saturation dissolved

in sea-water

Oxygen

Nitrogen

. i Carbon -r . i
^••g^" i dioxide 1 T^^^'

ml/l.|3oec

10-31
5-60

1811
10-74

0-54
0-30

0-51 29-47
0-20 j 16-84

In «/ / ^°^

35-0

33-2

61-5 1-8 1 1-7 100-0
63-8 i 1-8 1-2 1000

The solubility of gases in water is very strongly dependent on the temperature and
falls off rapidly as the temperature rises. The sea-water as a dilute salt solution shows
also a dependence on the salinity and the absorption coefficients fall with increasing
salinity. Fox (1905, 1907, 1909) has carried out extensive research on this subject, and
Rakestraw and Emmel (1937, 1938) have made further investigations. Table 25
shows saturation volumes at different temperatures and salinities for oxygen and
nitrogen. The weights present in mg can be obtained by multiplying the figures for
oxygen by 1-4292 and those for nitrogen by 1-2542. If oxygen-nitrogen ratios Oa/Ng
for different temperatures and salinities are worked out, it can be seen that there is
little variation; the dependence on salinity is small; with temperature it falls off
slightly.

For chemical methods of determining the oxygen and nitrogen contents of a sample
of sea-water see Report of "'Meteor' Expedition, 3 or "Oceanographic Instrumenta-
tion. Chemical Measurements" (Carrit, Nat. Acad. Sci. Nat. Res. Coun., no. 309,
pp. 166-85, 1952).

At the surface of the sea, in contact with the atmosphere, there is ample oxygen and
nitrogen available and it would be expected that the upper layers of the ocean were
saturated with both gases. This is generally the case, especially for nitrogen which is

The Sea-)vater ami its Physical and Chemical Properties

Table 25. Saturation values for oxygen and for nitrogen in sea-water in mill Hit res per
litre for a dry standard atmosphere

Temp.

Oxygen
salinity (%o)

Nitrogen
salinity (%„)

(. «-J

-2

9-50

916

8-82

8-47

812

901

8-68

8-36

803

7-71

1602

15-46

14-90

14-34

13-78

7-94

7-67

7-40

7-13

6-86

1408

13-62

1317

12-72

12-27

710

6-86

6-63

6-40

6-17

12-74

12-32

11-92

11-51

1110

6-43

6-23

604

5-84

5-64

11-57

11-20

10-84

10-48

1011

5-88

5-70

5-53

5-35

5-18

10-53

10-21

9-91

9-61

9-30

5-40

5-21

503

4-93

4-77

9-69

9-42

9-16

8-88

8-62

4-96

4-80

4-65

4-50

4-35

—

less reactive than oxygen and is biologically inert. Water samples from different depths
show mostly only minor deviations in nitrogen content from the saturation values.
This could be used to draw conclusions about the origin of deep layers and about the
vertical and horizontal displacements that they have undergone since their last
contact with the atmosphere, but since very few nitrogen determinations have been
made in the open ocean the method has so far been of little use. In any case, care
would be required in the interpretation of such results since super-saturation or in-
complete saturation may be due to other causes: to subsequent heating and cooling,
to the mixture of saturated water masses at diflTerent temperatures which always
leads to small super-saturation, to variations in atmospheric pressure and to the
production of nitrogen by bacteria that decomposes nitrite or nitrate. Since the equal-
ization of existing differences in saturation always proceeds slowly these deviations
will be conserved for a long time and can simulate water movements that would
otherwise be quite impossible.

The oxygen is also for the most part in equilibrium between the air and water at the
surface of the sea, but the deviations are more frequent and more marked than for
nitrogen. Besides the causes of more physical factors mentioned above (temperature
and pressure alterations, mixing, etc.), there are also biological factors which cause
variations. The respirations of plants and animals produces carbon dioxide and uses
up oxygen. Animals, however, obtain their essential carbon compounds from in-
gested organic material, while plants, on the other hand, obtain it by the assimilation
of carbon dioxide. This is converted with the help of sunlight into organic substances
and oxygen, which is set free, raising the oxygen content of the water.

The oxygen in sea-water is consumed not only by the respiration of living organisms
but also by the bacterial oxidation of dead organic matter and of organic compounds
in solution. This oxygen consumption is proportional to the rate of oxidation, which is
in the first place dependent on the temperature and also on the amount and nature of
the organic material present.

In the assimilation layer (the upper layer of the sea), usually down to the thermo-
cline (rapid density change in the vertical) conditions are rather complicated due to
the mutual interaction of the different factors. Oxygen is being steadily absorbed from

The Sea-water and its Physical and Chemical Properties 67

the atmosphere and produced by photosynthesis. Usually this addition is not exceeded
by removal of the respiration of the organisms present and by the oxidation of dead
material. Super-saturation by oxygen is thus quite possible and is occasionally found.
The surface layer is generally, however, the layer which is nearest to equilibrium with
the air. In the deeper layers of the ocean, below the assimilation layer, the oxygen is
provided almost exclusively by transport of the water from the surface by vertical
and horizontal movements. On the trajectories which the water particles perform
there is a continuous progressive consumption of oxygen so that the oxygen supply
in deeper layers depends either on the distance covered since the water mass left the
surface or on the speed with which it moved. A stationary state is only possible when
the supply of oxygen by renewal of the water mass and the oxygen consumption are in
equilibrium. Estimation of the oxygen distribution in the deeper layers of the ocean,
especially of the vertical and horizontal differences in saturation, until very recently
gave only the "age" of the water mass, i.e. the time since it left the surface layers.
After that some clarification had been obtained of conditions for similar processes in
lakes, the chemical-biological processes of oxygen depletion in the sea were further
elucidated by Seiwell (1937, 1938), Sverdrup (1938) and Wattenberg (1938). The
last one has discussed in detail the relevant chemical-biological factors in the ""Meteor'"'
Report and has pointed out its great importance for a proper understanding of the
distribution of oxygen in the ocean.

This distribution within the ocean shows that the explanation given can account
qualitatively for the oxygen producing and consuming factors mentioned above. The
maximum oxygen content is always found in the surface layers; in this skin layer mix-
ing by the wind and the waves and the turbulence due to ocean currents gives a more
or less even distribution that normally differs little from equilibrium with the atmos-
phere. The lower limit of this oxygen-rich layer, which coincides with the assimilation
layer, follows essentially the thermocline in the general oceanic structure. At this
transition layer, when it is strongly developed as is always the case in lower latitudes,
the oxygen content falls to a minimum. According to the geographical position of the
part of the ocean and the range of the annual convection at that point the depth of
this minimum varies between 100 and 1500 m. This oxygen-poor intermediate layer
is the most prominent feature of the oxygen distribution of the ocean in middle and low
latitudes. Below this minimum layer there is always oxygen-rich water with up to
70-90% saturation. As is explained later, this oxygen content of the deep-sea circula-
tion of the oceans originates from the major convection areas of the subpolar and polar
regions of the ocean where the water masses in the surface layers can sink to great
depths, and from there also fill the depths at middle and lower latitudes. In spite of
the long path travelled by these water masses there is little depletion because of the
low temperature and the small amount of organic material present, and the oxygen
content shows only a slight decrease. Figure 37 shows as an example the vertical distri-
bution of oxygen at about 10° S. in the South Atlantic; the vertical variation of density
is also shown and the density transition can be clearly seen. The right-hand side of the
figure shows the vertical changes in percentage oxygen saturation and in density a, at
a station in the North Atlantic near Greenland in the area where, according to a view
expressed by Nansen (1912), the North Atlantic deep water is formed and sinks
during the late autumn and winter. The almost constant value of the density down to

The Sea-water and its Physical and Chemical Properties

below 2000 m and the high oxygen content proves the possible presence of con-
vection descending to great depths and the considerable ventilation it would give.

The renewal of the deeper water layers has a major effect on the oxygen distribution
in them. If renewal did not occur oxygen depletion processes would in time reduce the
oxygen content until it would be finally zero. It is to be expected that enclosed, stag-
nating water masses will always have a very low oxygen content when their thermo-
haline structure prevents the thermal circulation from the surface reaching the bottom.
If the surface layer density is so low that it does not become heavy enough when the
temperature decreases in the autumn and winter in order to change places with the

6t 35 26
02% 20 I W

500

1000

1500

2000

2500

3000

Fig. 37. Left : Vertical distribution of oxygen and density at about 10° S. in the South Atlantic

(according to the values of the "Meteor" Expedition). Right: the same for "Meteor" station

122 (Greenland, ^ = 55° 3' N., A = 44° 46' W).

more saline deeper layers thus carrying oxygen to the layers beneath, the oxygen
content of the deep stagnating layers may fall to zero, especially when a lateral
addition of fresh water, due to the orographic conditions, is hindered or completely
missing. In this case hydrogen sulphide will be formed either by the decomposition of
proteins or by the reduction of sulphate by the carbon compounds of organic material
under the action of certain bacteria. The classic example of these conditions is the
Black Sea, where the water from about 200 m down to the greatest depths contains
considerable amounts of free hydrogen sulphide and thus forms a "Kingdom of the
Dead" from which all life has disappeared and where the organic world is represented
only by the lowest forms of plant life (Schokalski, 1924; Nikitin, 1927; Neumann,
1942, 1944). The thermo-haline structure of the Black Sea is indicated in Fig. 38 which
shows the vertical distribution of temperature, salinity, density, oxygen content and

The Sea-water and its Physical and Chemical Properties

hydrogen sulphide content at three stations in July in different places in the eastern
part of the Black Sea (Neumann, 1943). The station PM 298 lies in the southern
part, the station PM 308 lies in the northern part of the central eastern basin near an
area with little current, and the station PM 303 lies south-west of Sochum in the area
of the strong current along the Caucasian coast.

Hi^}°

t'7 9 II 13 15 17 19
5%«I7 IB 19 20 21 22 23

a-fW 12 13 14 15 16 17

7 cm'
2rc

Ks'Y

-^^ H,st

5 6 7cmyi

S%ol7

t7>l|

II 13 15 17 19 21 "C
19 20 21 22 23 %<.
13 14 6 16 17

ri 9 II 13 15 17 19 21°C

5%ol7 18 19 20 21 2 2 23 %o

o-,ll 12 13 14 15 16 17 18

100
200

400

600

800

1000

1200

1400-
1500

P.M. 298

42''00'IM

38°00'E

I6/I7-3ZII-1925

It
!\
li
j i
! i

P.M. 303

42°23'N
40°33'E
20-2Ii925

i !
it
il

ij
il

:i
— li-

^•^ 5%..'
H,S\.

P.M. 308

43''04-9N
38° 29-8'E
22-3ZIIi925

Fig. 38. Vertical structure of the water masses in the eastern part of the Black Sea. (Sept, 1925
stations: P.M. 298, 303, 308; temperature, salinity, density, oxygen content and sulphur

content.)

The vertical structure of the Black Sea is characterized by two layers. The upper
layer shows a very rapid increase of density with depth and usually extends down to
about 200 m. After a sharp bend in the a^-curve the density changes little with depth.
The boundary between these two layers coincides approximately with the upper limit
of the hydrogen sulphide; its depth varies from place to place depending on the dy-
namics of the currents prevailing. The upper layer (the troposphere) is divided in
summer at a depth of about 50-70 m by a definite temperature minimum at 6-5°-
7-5 °C. This surface zone has a constant salinity and shows a pronounced vertical
thermal convection; it is well ventilated and has a rich oxygen supply from the at-
mosphere and also from plant assimilatory activity. In the lower part of the surface
layer the oxygen falls off rapidly with depth and finally disappears, and in places is
replaced by hydrogen sulphide. The oxygen of these upper layers comes partly from
above and partly from horizontal advection but the latter effect is limited to the
immediate vicinity of the Bosphorus.

The whole of the layer from below the oxygen zone down to the bottom at about
2000 m has an almost constant temperature, about 8-8-9-0°C; the slight increase
from 300 m is largely an adiabatic effect. The principal characteristic of this lower
water is the hydrogen sulphide content which increases down to the bottom (see
Table 26).

Similar conditions, though on a smaller scale, are shown by several Norwegian
fiords where in most cases there is a considerable depth, a fresh-water influx at the

70 The Sea-water and its Physical and Chemical Properties

Table 26. Average vertical distribution of t, S and Hydrogen Stdphide in the Black Sea

Depth (m)

100

200

300

500

1000

1500

2000

Temp. {°C)
Potential temp. (°C)
Salinity (%«)

13-80

7-95

7-94

20-36

8-69

8-67

21-35

8-80

8-76

21-73

8-83

8-77

22-09

8-93 —

8-81 —

22-24 22-31

9-00

8-75
22-34

HjS content ml/1,
(standard pressure
and C)

0-0

0-45

1-42

3-45

5-55

6-09

6-24

inner end and access to the open ocean only over a bar or a very shallow sill. They have
recently been reviewed in detail by Munster (1936). Of 30 fiords on the western
and southern coasts, 16 showed hydrogen sulphide in the bottom layers; the other
14 had very low oxygen values varying between 0-22 and 5-47 ml/1. The ventila-
tion of the deeper layers depends in the first place on the sill depth and the width of
the passage to the free ocean. In some fiords changes in the hydrogen sulphide content
were found which must be due to the addition of ocean water.

The formation of hydrogen sulphide is only possible in closed or very poorly ven-
tilated deep basins. The Baltic which has a much lesser depth than the Black Sea
shows very similar hydrographic conditions, although the deep water in the Baltic
is renewed occasionally by the spasmodic entry of masses of North Sea water (Watten-
BERG, 1941) so that it is only in stagnant periods that the oxygen content is depleted by
the respiration of animals and by the oxidation of organic material in the water and
on the bottom. Table 27 shows typical conditions at a summer station in the middle of
the Baltic.

Table 27. Gotland Basin (57^24' N., 19°52' E.); ''Skagerrack''
Station Alf. 96, 31 July, 1922

Depth (m)

Temp.
(°C)

5%o

Density

Oxygen

Free
CO2

Ot '

ml/I. j %o satur.

ml/1.

15-52

6-64

4-19

6-78 101

0-23

10-60

7-27

4-37

7-64 103

0-28

415

7-36

5-91

— —

—

2-95

7-63

614

8-74 99

0 64

2-20

7-92

6-35

7-86 87

0-83

405

10-42

8-34

3-72

2-9

100

4-35

10-90

8-70 ,

3-62

2-7

150

4-55

12-59

10-04

3-72

2-4

209

4-60

12-81

10-21

2-45

3-3

The temperature minimum caused by winter cooling is at 60 m depth. The oxygen
content falls from the well-ventilated upper layer with 7-8 ml/1, to less than 2 ml/1,
a little above the bottom where it reaches only 30% of saturation. The carbon dioxide
content here is 3-3 ml/1., which is eight times the normal concentration (Schulz, 1924),

The Sea-water and its Physical and Chemical Properties

(b) The Carbon Dioxide Cycle in the Ocean and its Relationship to the Atmosphere

Unlike nitrogen and oxygen, the carbon dioxide in the sea is present not only in
solution but also in considerably larger amounts chemically combined as salts.
Conditions are thus much more complicated, and the situation has only been clarified
in recent times by Buch and McClendon using modern dissociation theory. Funda-
mentally one realized by this that the free and the combined carbon dioxide in solu-
tion are not independent of each other, but according to the law of mass action are in
chemical equilibrium with each other. The combinations occurring can be repre-
sented by the following equations :

CO2 (in the air) ^ CO2 (in solution )+ HgO ^ H2CO3 (carbonic acid).

The carbonic acid splits partially into its ions according to:

H2CO3 ^ H^

which can dissociate further by:

HCO3

(bicarbonation),

HCO3 ^ H+ + COg^ (carbonation).

All these forms derived from carbon dioxide are present in sea-water principally as
carbonate and bicarbonate ions, and only to a lesser extent in the free state. Equili-
brium exists between these forms, the carbonate and the bicarbonate ions, free carbon
dioxide and the hydrogen ion, and it will be discussed later. The reasons are now
understood for the long time needed in oceanographic research to obtain suitable
accuracy in the determination of the carbon dioxide in solution in sea-water.

Free carbon dioxide and carbon dioxide pressure. The solubility of carbon dioxide
in sea-water is relatively large, almost thirty times that of nitrogen. Fox investigated its
dependence on the temperature and on the chlorine content of NaCl solutions, and
corresponding measurements have been made by Krogh for sea-water. On this basis
of these investigations Buch and collaborators (1932) Wattenberg, (1936) prepared
tables showing the dependence of the solubility of carbon dioxide in sodium chloride
(NaCl) solutions on the temperature and the salinity. Table 28 shows a condensed
extract from these tables. The solubility of carbon dioxide decreases considerably
with increasing temperature and salinity. One litre of sea-water at 0° and 35T9%o S,
when in equilibrium with the atmosphere (partial pressure of carbon dioxide 0-0003

Table 28. Solubility of carbon dioxide in sodium chloride (NaCl) solution in
millilitres per litre at a carbon dioxide pressure of 1 atm.

1890

1713

1424

1194

1019

878

759

665

1708

1554

1291

1088

932

808

702

617

1622

1479

1228

1038

890

774

674

594

1605

1464

1215

1028

882

768

669

590

1588

1449

1203

1018

874

761

663

585

1572

1435

1191

1009

866

755

658

581

1540

1406

1167

990

850

742

648

572

72 The Sea-water and its Physical and Chemical Properties

atmosphere) contains 0-42 ml/1, of free carbon dioxide, which is very little. In water in
the uppermost layer of the open ocean the carbon dioxide content is usually not far
from the equilibrium value with the atmosphere.

According to Krogh's measurements in the North Atlantic the value for the carbon
dioxide pressure varies between 1-55 x 10^*and2-9 x 10~^. According to Brennecke's
values in the Weddell Sea ("Deutschland" Expedition) the carbon dioxide pressure
was higher than that in the atmosphere. Carbon dioxide in solution comes only slowly
to equilibrium with the atmosphere. Detailed investigations along this line have
been made by Buch (1917), in the waters around Finland, by Schulz (1923) in the
Baltic, by Wattenberg (1933) on the "Meteor" Expedition (1925-7 principally
between Africa and South America), by Deacon (1934) especially in the Arctic and
Antarctic regions and finally by Buch (1939, 1939^) in the North Atlantic and on a
cruise in the Arctic. All these measurements of the carbon dioxide pressure show
variations around the equilibrium position, sometimes the pressure in the water
is higher than in the atmosphere and at other times it is lower. These variations,,
however, are small as in the course of the long time which has been available, sea and
atmosphere have come into a mutual adjustment. Wattenberg (1936), from the ob-
servations available, arrived at the following conclusions (Fig. 39):

(1) There are limited areas of the sea where the carbon dioxide pressure of the water
is definitely greater than that of the air; these are principally places where rising water
currents bring water rich in carbon dioxide to the surface from intermediate layers
rich in carbon dioxide (west coasts of North and South Africa and of North and
South America).

(2) In other places there are, however, large areas where the carbon dioxide pressure
is somewhat less than the normal partial pressure of the atmosphere. These occur
especially in temperate and cold zones during the spring and summer, when rich plant
plankton is actively assimilating. There may be pronounced annual changes here in the
carbon dioxide pressure at the surface of the sea: a strong reduction in spring at
the beginning of diatom development and a gradual rise in autumn when dead
organisms start to decompose. See p. 77 for the distribution of carbon dioxide in
deep water and at the sea bottom.

Total carbonic acid. If the sea was neutral it would contain little carbon dioxide.
Sea-water is in fact alkaline and has a total carbon dioxide content that is much greater
than would be concluded from the carbon dioxide pressure. By far the largest part is
chemically combined in the sea salt.

The total amount of carbon dioxide present depends on the one hand on the car-
bon dioxide pressure and on the other on the amount of base available for combina-
tion with the carbon dioxide which is termed the alkalinity. Since the carbon dioxide
pressure is small, there is an almost linear relationship between the total amount of
carbon dioxide present and the alkalinity, and thus also the salinity since the
alkalinity is dependent very largely on this. Thus Buch (1914) for the Pojowick under
average conditions found the relationship

A = 0-07 + 1-00 CO, and COg = 0-32 - 0-1735'

where CO, is expressed in millimoles/litre and A in milliequivalents. Similar relation-
ships were also derived for the Gulf of Finland and the Gulf of Bothnia.

The Seo-water and its Physical and Chemical Properties

Fig. 39. Distribution of carbon dioxide pressure (given in 10-* atm.) at the surface of the
South Atlantic (according to Wattenberg).

In the open ocean the average value for total free carbon dioxide is usually between
45 and 55 ml/1. Ruppin has found for the middle North Sea 45-9 for the Beltsea 36-7
and for the southern part of the Baltic 31-9, while Brennecke (1909) found values be-
tween 46 and 55 in the Atlantic and in the Indian Ocean and between 45 and 59 in
the Antarctic Ocean. In the North Sea Knudsen (1899, "Ingolf" Expedition) found
lower values, between 34-1 and 46-6 ml/1.

Alkalinity. In sea-water the sum of the cations of bases (Na+, K+, Mg2+, Ca2+), is
always a little greater than the sum of the anions of strong acids (SO^-, CI", Br-).
This excess of base is known as the "alkaline reserve"; it gives sea-water an alkaline

74 The Sea-water and its Physical and Chemical Properties

reaction and is very largely present in the form of carbonates and bicarbonates. Since
Tornoe, it is also known as the "alkalinity", a term which is also used for the hydrogen-
ion concentration. To avoid confusion the sum of the carbonate and bicarbonate ions
is termed in oceanography (following Buch) the "titration alkalinity". This is ex-
pressed in the equation

A = 2[C02-] + [HCO3],

and can be found by simple titration with hydrochloric acid (Wattenberg, 1933; see
also 1930).

The alkaline reserve in sea-water is largely combined with carbonic acid, but a
smaller part is also combined with other acids the most important of which is boric
acid. Sea-water of 35%o S contains 4-7 mg/1. of boric acid (Buch, 1933). The last
anomalies in the carbon dioxide system of sea-water have only been eliminated by
taking this acid into consideration since it and its ions are definitely concerned in the
equilibrium despite their small concentration.

Since the individual constituents of the salt in sea-water are in almost constant
ratio to one another, it would be expected that the amount of base available for the
formation of carbonate and bicarbonate, that is the titration alkalinity, would be
directly dependent on the salinity. This is the case. The dependence between the two
was first shown by Hamberg (1885) and the investigation by Brennecke of the surface
samples collected on the "Deutschland" Expedition gave the relationship between
them as A = 0-06119S (according to Schulz, 1921). Later investigations have shown
that for the open ocean the dependence of alkalinity on the salinity is given with
suflftcient accuracy by the relationship

A = 0-068S%o = 0-123 CI (in milliequivalents).

This simple proportionality does not apply to the sea-water of the marginal and ad-
jacent seas as has been shown by Ruppin and Buch; these variations appear to be due
to the inflow of fresh water from the land. The North Sea and the Baltic, especially in
coastal areas, show alkalinity values that are higher than would correspond to the
salinity (addition of carbonate in river water). Similar conditions are found in the
Gulf of Bothnia, the Gulf of Finland and in the Adriatic.

Carbonate at the sea bottom passing into solution has the same effect as the addi-
tion of carbonate from the land. The investigations of the "Challenger" Expedition
clearly indicated that the water immediately above the sea bottom was more alkaline
than that at the surface or in the middle layers (Dittmar, 1884; Brennecke, 1921).
The more accurate alkalinity determinations of the "Meteor" Expedition 1925-7
showed definitely that the specific alkalinity (the ratio of alkalinity to chlorinity,
A : CI) almost always increased near the sea bottom. This increase can only be ex-
plained by calcium carbonate from the bottom sediments going into solution (see
p. 85).

Hydrogen-ion concentration. Pure water dissociates according to the equation

HoO ^ H+ + OH-.

The Sea-water and its Physical and Chemical Properties 75

H+ is the positively ciiarged hydrogen ion and OH~ is the negatively charged hydro xyl
ion. The law of mass action gives the equation

[H+] • [OH] _
[H,0] '''"'

where the square brackets indicate concentrations in mols per litre. The concentration
of pure water [H2O] is approximately the same for all dilute aqueous solutions such as
sea-water. Since [HgO] is constant for a given temperature it can be included with the
constant K^^ so that

[H+] • [OH-] = K,,.

At 18^ 25° and 50°C K,, has the values 0-61 x 10-", 1-0 x 10-^^ and 5-4 x 10-^*
respectively. The concentration of either of the ions can be calculated if that of the
other is known. Solutions where [H+] > [OH"] are acid and where [H+] < [OH-]
are alkaline; in neutral solutions the two concentrations are equal. The character
of the solution is thus specified completely by [H+]. In pure neutral water at 25°C
[H+] = [OH"] = VK^ = 10"'^. The hydrogen ion concentration of a solution is
usually not given as [H+] but as the quantity —log [H+] = pH. For pure water at
25 °C the pH is thus 7-0.

Carbon dioxide system in sea-water. There is an equilibrium between the different
chemical species derived from carbon dioxide that are present in sea-water and this
must follow the law of mass action. As for every electrolyte there is a reciprocal re-
lationship between the concentrations of the undissociated substance and those of its
ions. For the first and second dissociations of carbonic acid

[H^l • [HCO;l ^ i^y^^^K,.

[H2CO3) ' [HCO;l

To these equations can be added the equation for the titration alkalinity

2[C02-] + [HCO3] = A.

Since the dissociation constants Ky and Ko are known, these three equations contain
four unknown quantities

[H+]; [HCO;]; [CO^-] and [H^COg].

If one of these can be determined, for instance the pH = (—log [H+]) then the other
three can be calculated.
The dissociation constants for carbonic acid in pure water (18°C) are

Ky = 3-06 x 10"' and K^ = 5 x 10"".

In sea-water the values of these dissociation constants are different because of the effect
of the considerable amounts of other ions present in sea-water. The ions of the neutral
salts such as Na+, K+, Mg2+, S0^~ also affect the carbon dioxide equilibrium but not

77?^ Sea-water and its Physical and Chemical Properties

in proportion to the total amount present: according to the theory of interionic forces
developed by Milner, Bjerrum, Debye and Huckel, amongst others, only a small frac-
tion is involved. This fraction of the total concentration is termed the "activity"; the
equilibrium thus involves not the total concentrations of the different ions, for instance
[H+] but the activities, in this case/JH+J, where/is the "activity coefficient" and the
above equations are replaced by others where the factors on the left-hand side are
multiplied by the activity coefficients /i, /a, /g and/4. The constants Ki and K2 remain
unchanged; they are termed "activity constants". However, instead of taking the
effect of the neutral salts directly into consideration it can be allowed for by its effect
on the dissociation constant; the apparent dissociation constants K[ and Kl are termed
the "concentration constants". At the suggestion of the International Council for
Oceanography Research they have been determined by Buch and co-workers (1932)
Wattenberg, (1936). Table 29 gives numerical values for —log K[ and —log K2 for
different temperatures and salinities (see also Buch, 1951).

The calculation of the concentration of the individual forms of carbon dioxide in
sea-water (free carbon dioxide, carbon dioxide pressure, carbonate and bicarbonate

Table 29. Values of the first and second dissociation constants of carbonic acid in sea-
water at different temperatures and salinities

-\ogK[

log a:;

S%o

0°C

10°C

20°C

30°C

0°C

10°C

20°C

30 °C

6-66

6-57

6-49

6-43

10-56

10-45

10-34

6-32

6-23

616

611

9-59

9-46

9-35

9-24

6-29

620

612

607

9-47

9-34

9-23

912

6-23

614

606

600

9-32

9-20

909

8-98

619

610

602

5-95

9-24

912

9 00

8-80

ions and total carbon dioxide) is now a simple calculation if the hydrogen-ion con-
centration, the pH, is measured directly and the titration alkalinity is found from the
salinity using the relationship given on page 74. This calculation can be shortened
considerably if the carbon dioxide pressure and the total carbon dioxide are tabulated
or plotted graphically for the most frequently occurring values of salinity, temperature
and pH.

The relationship between pH and the concentrations of free carbon dioxide,
carbonate and bicarbonate can be shown clearly in a diagram (Fig. 40), where the
percentage of each form is given as a function of the hydrogen-ion concentration. The
S-shaped curves separate these factors in such a way that for any value of the pH
the composition of the total carbon dioxide present is given along the ordinate. The
curves for sea-water are drawn with full lines, the curves for pure water with dashed
lines; the first is displaced towards lower pH-values. The presence of neutral salts in
sea-water displaces the equilibrium towards the acid side because the apparent dissocia-
tion constant increases. It can be seen that at very low pH-values there is almost only
free carbon dioxide present, as the pH rises the concentration of bicarbonate increases
and reaches a maximum at pH = 7-5; the carbonate ion becomes important only at
higher pH-values, The two vertical lines in Fig. 40 show the normal range of the pH

The Sea-water and its Physical and Chemical Properties

in the open ocean. It comes within the range where all three factors: HCO3, COg" and
free CO2 are present in measurable amounts, although bicarbonate predominates
considerably.

The above values for the apparent dissociation constants are for water at a pressure
of one atmosphere. If the pressure is increased the constant also increases since the
pressure strengthens the dissociation both of the carbon dioxide and of the neutral

Fig. 40. Percentage distribution of the three forms of carbon dioxide (free carbon dioxide,
bicarbonate, carbonate) in pure water and in sea water as a function of pH (according to

Buch).

salts. This dependence implies, as shown in Table 30, that water displaced from the
surface downwards to great depths will be more acidic, and inversely that of a sample
brought from a definite depth with a collecting bottle will as a consequence of the
decrease of pressure show a higher pH (be more alkaline).

Table 30. Dependence of the concentration constants for carbon dioxide
CO2 on the hydrostatic pressure

Depth in m

2000

4000

6000

8000

10,000

10
10

1-26
110

1-58
1-20

200
1-32

2-45
1-41

i 3-02x^nat
' 1-55 X K'.y'^

Wattenberg gives the example shown in Table 3 1 of this effect. This pressure effect
has a practical significance in processes involving the hydrogen-ion concentration such
as the life of deep-sea organisms and the solubility of calcium carbonate at the sea
bottom.

Carbon dioxide in the deep layers of the ocean. The work of the "Meteor" Expedition
1925-7 gave the first reasonably good information on the distribution of carbon di-
oxide in the deep layers of the sea. The essential results have been summarized by
Wattenberg (1936). For the most part there is an approximate equilibrium at the
surface of the sea between the partial pressures of carbon dioxide in the sea and in the

The Sea-w'oter ami its Physical and Chemical Properties

Table 31. Variations of pH with depth at constant carbon dioxide
content due to the change in pressure
(After Wattenberg, 1936 )

Depth (m)

2000

4000

6000

8000

10,000

7-80

7-75

7-70

7-65

7-60

7-55

>■ 800

7-95

7-91

7-87

7-82

7-78

8-20

816

812

808

804

8 00

atmosphere (see p. 72). Down to 50 m there is a slight reduction in the carbon dioxide
content due to the assimilatory activities of the phytoplankton. Then follows a thin
layer where the effects of assimilation and respiration are in balance. Beneath this
layer the carbon dioxide pressure rises until it reaches a pronounced maximum at a
depth of 500-1500 m (intermediate layer) depending on the latitude; it then falls off
again, at first steeply and finally in the deeper layers approaches the values found at
the surface. This carbon dioxide inversion (see Fig. 41) is accompanied by a change in
the pH which is almost the exact mirror image. Figure 42, which shows the carbon di-
oxide pressures along a cross-section through the subtropical South Atlantic, illus-
trates how clearly marked these changes are. According to Wattenberg, these pro-
nounced variations in the carbon dioxide distribution are due principally to the follow-
ing factors :

( 1 ) The strong renewal of the deep water of the oceanic stratosphere by water masses
of polar and subpolar origin which sink during the late autumn and winter in

7-6

78 80 8-2

,-'

-■

— ^

H'\

^.y

1000

i '

2000

\
i

1
1

5000

:o.

1
i
i
i

\/
A

jpH

4 6 8

CO,- pressure

Fig. 41. Vertical distribution of the carbon dioxide pressure ^002 CO"* atm), the hydrogen-
ion concentration pH and the temperature in middle latitudes of the Atlantic (according to

Wattenberg).

The Sea-water and its Physical and Chemical Properties

lOOO

2000

300

4000

5000

600O

Fig. 42. Carbon dioxide pressure cross-section through the subtropical part of the South
Atlantic (8-5 -13' S., profile VIII from the "Meteor" Expedition; given in 10"* atrri).

higher latitudes and reduce the carbon dioxide content of the water at middle depths
(2000^000 m).

(2) The decomposition of dead organisms that takes place principally in the upper
layers beneath the transition layer. In shallow seas dead organisms reach the bottom
before decomposition is complete and the carbon dioxide pressure thus increases down
to this depth. In the deeper layers of the major oceans decomposition occurs largely in
the upper layers and the carbon dioxide pressure then decreases with further increase
in depth.

(3) The respiration and oxidation processes that produce carbon dioxide proceed
more rapidly at the higher temperatures in shallow depths than at greater depths where
the temperature is lower.

All three factors combine to bring about the observed distribution, although a sta-
tionary state can naturally only occur when the addition and the consumption of
carbon dioxide are in equilibrium. However, for quantitative considerations of this
type there is as yet no numerical estimate of the effect of the different processes.

In the last hundred metres immediately above the sea bottom there is a more or less
large increase in the carbon dioxide content above the almost constant value of the

The Sea-water and its Physical and Chemical Properties

deeper layers (see Fig. 41). This apparently almost universal phenomenon may be due
partly to the slower renewal of the water in the layer next to the bottom and partly
to the gradual decomposition of material, not easily oxidizable, which with the shells
and skeletal parts of organisms forms the sediments of the bottom and makes possible
the formation of carbon dioxide in the bottom layer. This bottom layer with a definite
increase is particularly well developed and sharply separated from the upper layers in
the western half of the South Atlantic in the area of Antarctic bottom water (see Fig.
43).

The carbon dioxide system between the ocean and the atmosphere (BuCH, 1942).
The state of equilibrium at the surface of the sea between the ocean and the atmosphere

9(r 80" 70" 6(r 50" <tO" 30* 20* 10° 0' 10* ZtT ZV hV 50* E

Fig. 43. Distribution of carbon dioxide pressure (10"^ atm) at the sea bottom (below
4000 m) (according to Wattenberg).

77?^ Sea-nater and its Physical and Chemical Properties 81

does not extend over the whole surface. More recent investigations have shown that
measurable variations occur, though they tend towards equilibrium. To investigate
more closely the direction of variations in the carbon dioxide content of the atmosphere
from equilibrium with that of the sea, and the mutual interaction of the two, it is
necessary to know: (1) the nature of the factors causing changes in the carbon dioxide
content in both spaces; (2) the distribution of the carbon dioxide in both media when
equilibrium has been finally established; and (3) the duration of the exchange process
leading to a new equilibrium and, dependent on that, the extent to which the sea and
the atmosphere come into contact enabling equalization of the differences between
them.

As far as the first point is concerned, the principal source of the changes in the
carbon dioxide content appear to lie in the atmosphere. Goldschmidt (1934) has
given a general carbon dioxide budget for the atmosphere and the sea which is of
fundamental importance for the present problem. Table 32 shows the amounts of
carbon dioxide in y (=0-001 mg) per cm^ of the total surface of the Earth entering
or leaving the atmosphere and the sea annually. "Juvenile" carbon dioxide enters
the atmosphere from volcanoes, fumaroles and carbonated spring water. The value
given in Table 32 is the order of magnitude of the steady supply that would give the
total amount released during the course of geological history. In more recent times
there has been a particularly large increase in the amount of carbon dioxide entering
the atmosphere due to the steadily growing combustion of coal and oil by man.

Compared with this large addition of carbon dioxide the amount removed from the
cycle by weathering processes and by the formation of carboniferous sediments is
very small. All these processes are, however, greatly exceeded by the amounts of
carbon dioxide involved in the biological processes of assimilation and respiration.
These two processes appear very largely to balance each other. The combustion of
coal by man can, however, as shown in Table 32, produce in time a measurable change
in the carbon dioxide contents of the atmosphere and the ocean, in spite of its small
annual effectiveness.

Table 32. Annual budget of carbon dioxide per square centimetre of the

Earth's surface
(After Goldschmidt, 1934)

f juvenile COg 3-6 y

Supply by -^ industrial combustion of coal and oil 800 y

l^respiration and decomposition Approx. 40,000 y

r photosynthesis Approx. 40,000 y

Consumption by ^ weathering processes 3-4 y

l^the formation of carboniferous sediments 0-3-2 y

The addition of 0-0008 g/cm^ over a period of 35 years (1900-35) would give an
increase in the carbon dioxide content of the atmosphere of 0-028 g/cm- provided that
all this carbon dioxide remained in the atmosphere. A more recent and somewhat
more detailed presentation of the carbon dioxide cycle in the atmosphere, the hydro-
sphere and the lithosphere has been given by Lettau (1954) and is shown in Fig. 43a.
This gives detailed information on the individual parts of these interchanges and shows

The Sea-water and its Physical and Chemical Properties

that compared with the long cycles of water HgO, there are only short vertical cycles.
This is a consequence of the non-existence of a liquid carbon dioxide phase. Accord-
ing to Rankama and Sahama (1950) the total mass of the carbon dioxide in the
atmosphere is 23 x 10^^^ g.

It should be emphasized that the existence of a pressure difference in the carbon
dioxide cycle between the atmosphere and the ocean will always lead to an interchange

ATMOSPHERE

+ 3+4 -1

A A

+ 2

-2

t -3 -4+29-28 i--^-2^^

XmDRt

LJTHOSPHEfiE

sssg

Eifi^ftJE^

Fig. 43o. Schematic diagram of the carbon dioxide cycles in the atmosphere, the hydro-
sphere and the lithosphere. 100 relative units = 16 ■: 10^^ g CO2 per year or 0032 g cm"^
years"^ Note that biological processes are dominant, particularly those of marine life.
Balances: atmosphere + 9 — 5 = 4; lithosphere -y- 29 — 35 = —6; hydrosphere
- 62 - 60 = 2: total -r 100 - 100 = 0. The atmosphere gains 004 : 16 x lO^^ = 0-64
X 10'^ g per year or 3-2 x 10'" g in 50 years which corresponds to 14% of the estimated
total CO2 amount of 23 : 10^^ g present in the entire atmosphere.

->• , Release from rocks; •

ooooo>. , Forest and prairie fires ;
/^^ , Respiration; >

tension differences in the sea;

► ••••»- , Deposition in sediments and minerals;
• ••••»> , Combustion of coal and oil ;

, Assimilation; »- , Flux following CO2

f = value smaller than 0-5 rel units
(Lettau, 1954).

between the two media that will cease only when equilibrium is established. Schlosing,
in laboratory investigations, has clarified these exchange phenomena and shown that
the sea always has a levelling effect on pressure differences that occur between the
atmosphere and the sea. Since the sea has a carbon dioxide content several times
greater than that of the atmosphere it suppresses fluctuations in the atmospheric carbon
dioxide content and it tends to hold the atmospheric carbon dioxide at a constant
value. In this respect the sea acts as a "regulator" of the carbon dioxide content,
opposing changes in the content of this gas in the atmosphere. Nevertheless, recent
investigations have shown that the variations in carbon dioxide content in both the
ocean and the atmosphere are of the same order of magnitude. The sea in acting as a
damper thus undergoes the same variations as the atmosphere, and under these
conditions it is not easy to decide which is the "regulator" and which is the passive
part. When changes occur and a new equilibrium is established, the sea of course takes
up a much larger amount of carbon dioxide than the atmosphere. According to Table
32 the annual production of carbon dioxide amounts to about 0-0008 g/cm^ of the
Earth's surface. The amount of carbon dioxide already present in the atmosphere
amounts to about 0-4 g/cm^. If the whole of the carbon dioxide produced remained

The Sea-water and its Physical and Chemical Properties 83

in the atmosphere the present carbon dioxide content would be doubled in 500 years.
In actual fact if there is a pressure difference between the ocean and the atmosphere
the sea takes up carbon dioxide until this difference vanishes. Buch (1939) has cal-
culated that if the ocean and the atmosphere are always in equilibrium then five-sixths
of the carbon dioxide produced is absorbed by the sea while only one-sixth remains
finally in the atmosphere. Thus, if the sea absorbs the industrial carbon dioxide so
rapidly that equilibrium is always maintained, then its present content would double
at first in 3000 years.

However, some time is needed to reach a new equilibrium and this is probably not
reached as quickly as is customarily assumed. The cause could lie in the very slow
vertical circulation within the ocean. In a short time only a very thin contact layer can
interchange with the atmosphere. The equihbrium time for the whole volume of the
ocean should certainly be more than several thousand years, and it must also be
remembered that the initial pressure differences are very small and at first rise only
slowly. According to the investigations of Buch in the North Atlantic in summer 1935
and in the sub-arctic regions in summer 1936, this part of the ocean and of course the
corresponding region in the Southern Hemisphere appear to be the only areas where
over long periods carbon dioxide is absorbed from the air in water masses which, by
convective sinking in the autumn and winter, convey it to the rest of the ocean. Only in
these layers is a rapid renewal of the surface water to be expected and these are thus
the principal sites of equilibration in the carbon dioxide interchange between the ocean
and the atmosphere (see also Buch, 1948).

At the present time insight into the dynamics of these processes is rather inade-
quate due to the scarcity of carbon dioxide pressure determinations. Extensive syste-
matically collected series observations are needed for a better understanding of these
phenomena. A more accurate investigation of the distribution of carbon dioxide in an
adjacent sea (the Baltic) has been described by Buch (1945).

The solubility of calcium carbonate in water increases with the carbon dioxide
content. This can be explained chemically as follows: calcium carbonate in solution
is almost completely dissociated into Ca^^ and C0|" ions according to the equation

CaCOg ^ [Ca2+] + [CO^-].

Since the concentration of undissociated calcium carbonate is very small and, if the
sea-water is saturated, must be constant, the solubility product is given in a first ap-
proximation by

[Ca2+] . [CO^-] = Ki

In the carbon dioxide equilibrium shown on p. 75 most of the hydrogen ions present
combine with the carbonate ions to form bicarbonate ions since the bicarbonate ion,
HCO^, is dissociated only to a small extent. This alters the calcium carbonate equili-
brium, and calcium carbonate will thus go into solution until [Ca^+] increases suffi-
ciently to satisfy the equilibrium equation. The equilibrium thus depends on the con-
centrations of all the ions, H+, HCOg", CO^- and Ca^^ involved (Wattenberg, 1933,
1936).

The Sea-water and its Physical and Chemical Properties

As well as the concentration of free carbon dioxide there are other factors also that
affect the solubility and, while they are not so important, they must still be taken into
consideration. The first of these is the concentration of Ca2+ derived not from the
dissolved calcium carbonate but from the calcium sulphate and calcium chloride, that
is, the excess of calcium ions above that corresponding to the combined carbon
dioxide. These calcium ions, by the law of mass action, reduce the solubility of the
calcium carbonate. An additional factor affecting the situation is the increase in the
solubility product due to the presence of neutral salts in the same way as for carbon
dioxide. Table 33 shows that the constant K^ is a hundred times greater in sea-water
than in pure water.

The solubility constant depends not only on the salinity but also on the temperature
and, unlike most salts, decreases with increasing temperature. The pressure (at
constant carbon dioxide pressure) also has a considerable effect on the solubility of
calcium carbonate, but it is not yet certain how large this effect is. The factors affecting
the solubihty of calcium carbonate in sea- water thus fall into two groups: (1) those that
increase the solubility such as increasing carbon dioxide concentration, salinity and
hydrostatic pressure ; (2) those that decrease the solubility such as increasing tempera-
ture and calcium concentration. The values for solubility given in Table 34 show that
in sea-water these factors more or less compensate each other so that there are no
major differences from the solubility in pure water.

Table 33. Dependence of the solubility product, K'^,for calcium carbonate on the salinity

and temperature
(After Wattenberg, 1936)

5 in %„ (at 20^C)

Temp, in °C (at 35 %„ S)

10 25 ! 35

' 0

10 20 30

1 0-5

22 j 48 62 X 10-**

j 8.3

7-4 6-2 1 4-4 X 10-'

The calcium carbonate content can be found by determination of the alkahnity which
varies in direct proportion to the variations (in milliequivalents/litre) in calcium car-
bonate. Since the variations in calcium content are not very large it is necessary to
determine the alkalinity very carefully (Wattenberg, 1930).

Table 34. Solubility of calcium carbonate
(CaCOg) in milligrams per litre in
sea-water (355'%o)/or different tempera-
tures and carbon dioxide pressures (in
10-4 atm)
(After Wattenberg, 1936)

Pco2 X 10*

t°C

1 2

100

135

105

The Sea-)\'ater and its Physical and Chemical Properties

The mean vertical distribution of calcium carbonate in the Atlantic between
20° N. and 20" S. is shown in Fig. 44 from the results of 236 determinations made
during the "Meteor" Expedition. The variations in calcium carbonate content can be
divided into two groups: (1) Those due to differences in the total salinity — the alka-
linity and therefore the calcium carbonate both increase with increasing salinity;

CaCojmg/l.

CoCojg/kg Solt
3-32 3-36 3-40 3 44 348 352

1000

2000

^ 3000

4000

5000

Fig. 44. Mean vertical distribution of salinity (S^/q^, calcium carbonate (mg per litre of
water) and calcium hydroxide (in g per kg of salts). The last curve gives a measure of the
deviation of the calcium hydroxide content from proportionality with the salt respective to

chlorine contents.

(2) Those caused by chemical and biological changes. To show the last more clearly the
calcium content is given not in terms of unit volume of water but in unit weight of
salt; these are therefore expressed in g CaCOg per kg of salt or as per mill (%o). This
then shows the variations in the calcium carbonate content of sea- water from propor-
tionality with the salinity or the chlorinity. These are particularly important; they are
furthest from normal at two places: (1) in the surface layers close to the atmosphere;
(2) in the layer immediately above the sea bottom. The relatively small calcium car-
bonate content of the very surface layer must be attributed to consumption by plank-
ton, while the sharp increase at the sea bottom must be due to calcium carbonate
dissolving from the sediments at the bottom.

The normal calcium content of the water in the open ocean can be taken as about
3-40 g CaCOg per kg of salt. The depletion of calcium carbonate in the surface layer
is about 2% and the enrichment at about 50 m above the bottom is about 4%.
Special measurements would be needed to determine whether there is ii further in-
crease nearer the bottom. The maximum value in the bottom water is not the same
everywhere. It appears to be larger the greater the depth of the bottom, as is shown in
Table 35 (Wattenberg, 1931).

At depths of 3000 m this increase begins at about 300 m above the bottom, at
4000 m depths at 600 m and at 5000 m depths at about 1000 m. There are two fac-
tors involved in producing this apparently stationary state: (1) the continuous upward
flux of the calcium carbonate content as specified above ; and (2) the advective transport

The Sea-water and its Physical and Chemical Properties
Table 35

Bottom depth in metres 2000

3000

4000 5000

CaCOg-content in g/kg salt of the
deep water (at 50 m above the
bottom)

3-398

3-448

3-500 3-525

of more or less calciferous water by the bottom currents. Approximate calculations
made by Wattenberg (1935) have given an intensity for bottom currents which agrees
well with that deduced from oceanographic factors.

The surface and bottom waters show regional differences, while the intermediate
water masses of the ocean show practically the same calcium carbonate content

90" 80° 70° 60° 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50°

Fig. 45. Chart of the calcium percentage saturation of the surface water in the Atlantic Ocean
(CaCOa) (according to Wattenberg).

The Sea-water and its Physical and Chemical Properties 87

throughout. In the polar and subpolar regions the surface minimum is largely absent
because of the winter convection which eliminates the minor depletion of calcium
carbonate by the few calcium-carbonate-consuming organisms during the brief summer.
In low latitudes, on the other hand, the depletion of calcium carbonate is particularly
pronounced due to the isolation of the upper layer by the thermocline in the tropics.
The regional differences in the bottom layers are shown principally by the degree of
saturation with calcium carbonate.

The degree of saturation of sea-water by calcium carbonate in solution is found by
comparison of the actual content with the solubility of the water in situ. Calculation
of the degree of saturation shows that the surface water in equilibrium with the atmos-
phere is supersaturated with calcium carbonate at all temperatures found. In the
tropics and the subtropics the supersaturation is very large and the water may contain
up to three times more calcium carbonate than that corresponding to the equilibrium
value; Fig. 45 shows the percentage saturation with calcium carbonate in the surface
water of the Atlantic. The large supersaturation in low latitudes shows very clearly;
only the presence of this supersaturation allows an equilibrium between the addition
of calcium in river water and its consumption by various organisms and sometimes by
spontaneous inorganic precipitation at the bottom. It can be readily understood that
the production of calcium by various organisms is facilitated and favoured by this
supersaturation.

Beneath the thermocline in low latitudes the saturation value falls rapidly with in-
creasing carbon dioxide pressure to below 100% and reaches a minimum in the inter-
mediate layers; in places the saturation may fall to less than 92%. While there are no
large differences in the deep water below 1500 m (degree of saturation 98-100%)
the bottom water in the Atlantic Ocean differs somewhat in saturation just as it also
differs in carbon dioxide content.

Calcium is also involved in a closed cycle. In the upper layers of the sea there is a
strong withdrawal of calcium carbonate, partly by biological processes associated
with calcium using animals and plants and partly by inorganic precipitation.

Compensation is performed at the sea surface by the supply of calcium due to river
water and at the sea bottom by solution from the bottom sediments. Without an
accurate quantitative estimation of the individual components in the cycle it is im-
possible to state whether supply or consumption predominates, or whether the present
condition of considerable supersaturation at the surface of the sea is a stationary state.

Chapter III

Temperature in the Ocean ^ the Three-
dimensional Temperature Distribution
and its Variation in Time

1. Heat Sources, Heat Exchange and Heat Budget in the Ocean

All changes of state in the liquid and gaseous envelopes of the Earth are due basically
to energy changes. Energy is very largely supplied from outside the Earth, principally
from the sun which provides an inexhaustible source of radiant energy for the Earth.
There is a constant inflow of energy from the sun and a constant outgoing radiation
from the Earth into space. The Earth does not retain the energy supplied to it but
returns all except a vanishingly small part to outer space in the same form (radiation)
in which it received it. The possibility of life on the Earth and all changes of state on
the Earth depend not so much on the inflow of solar energy as on the enormous supply
of entropy involved in the conversion of the high-temperature radiation from the sun
to the low-temperature radiation from the Earth.

These considerations lead to the concept of a stationary state as far as the heat
energy of the Earth, taken as a whole, is concerned. This constancy in heat energy can
be confirmed for the solid part of the Earth and for the atmosphere, and it can be
expected that it holds as a close approximation for the energy budget of the oceans.
There are, of course, small variations with time in the temperature of the ocean, but
these can be taken as variations around a mean value which remains essentially un-
changed .

Heat budget of the ocean. Tn this quasi-stationary state all the supply in energy is
balanced by equally large losses of energy. The most important factors are the radia-
tion, the interchange of sensible heat with the atmosphere above the sea and evapora-
tion from the surface of the sea or the condensation of atmospheric water vapour.
Other minor sources of heat that may be mentioned besides the above stated ones are
listed in Table 36.

The order of magnitude of the heat amounts involved in each of these processes
varies considerably. The largest is certainly the heat absorbed from solar and sky
radiation which is the principal factor in the heat budget of the very upper layers of the
sea. At its upper limit the Earth atmosphere obtains per cm^ by normal incidence an
energy of l-94g cal/min (solar constant). The entire surface of the Earth receives
per cm^ on the average 0-485 g cal/min or during the entire day 700 g cal/cm^. This
incoming radiation from the sun is largely short wave. Its intensity is decreased on
passing through the atmosphere so that only 43%, that is 0-21 g cal cm - min"^

The Three-dimensional Temperature Distribution and its Variation in Time 89

Table 36.

Heat sources Heat losses

1. Absorption of solar and sky radiation 1. Radiation from the sea surface Qb

2. Convection of sensible heat from atmos- 2. Convection of sensible heat from sea to
phere to sea atmosphere Q/,

3. Conduction of heat through the sea bot- 3. Evaporation from the sea surface Q^.
tom from the interior of the earth

4. Conversion of kinetic energy into heat

5. Heat produced by chemical and bio-
logical processes

6. Condensation of water vapour on the
sea surface

7. Radioactive disintegration in the sea-
water

reaches the surface of the sea. Of this, 27% is direct solar radiation and 16'^o is diffuse
radiation from the sky (sky light). From the other sources listed in Table 36 those with
a comparatively smaller effectiveness can be neglected. The sources listed under item
2 for heat gain and loss can be added giving one source. The same applies to item 6
(heat gain) and item 3 (heat loss).

The heat obtained from the interior of the Earth is about 50-80 g cal/cm^ per year
or on the average about 10 x 10"^ g cal/cm^, min. The heat supplied from the in-
terior of the Earth has recently been measured directly for the deep-sea basins of the
Pacific Ocean by Revelle and Maxwell (1952) and for the Atlantic Ocean by
BuLLARD (1954). These measurements gave in agreement the value 6-2 x 10~' g
cal/cm- min which corresponds to the value for the continents. Compared with the
heat from solar radiation this is unimportant; it probably causes only small local
variations in the thermal structure of deeper, enclosed stagnating water (see Chap. III.
4d).

The kinetic energy which the sea obtains by the tangential action of the wind on the
sea surface and by the dissipation of tidal energy by turbulent friction and which will
be transformed into heat gives only a very small heat contribution. The energy im-
parted by the winds amounts scarcely to a ten-thousandth part of the solar and sky
radiation energy and can therefore be neglected. Also the tidal energy dissipated by
turbulence is only of any appreciable influence in shallow waters. For example, Taylor
found the value of 1050 g cal/cm^ per year = 0-002 g cal/cm^ min for the Irish Sea
(see Vol, U. Chap. XV, 3). If this heat could accumulate in the Irish Sea for a whole
year the temperature rise would be only 0-2 °C. The item 5 in Table 36 has no significance
in the general budget of the sea and only requires to be taken into consideration where
there are local concentrations of plant life. The disintegration of radioactive material
in sea-water will afford barely 4 x lO'^gcal/cm^ min. Under these conditions the
heat budget of the ocean requires the following equation

Qs - Qb- Qh ~ Qe = 0.

90 The Three-dimensional Temperature Distribution and its Variation in Time

For particular parts of the sea and for short intervals of time it may also be necessary
to take into consideration the heat carried by ocean currents, or by mixing processes
into or out of the oceanic region under consideration and also the heat which causes
over short periods of time changes in water temperature. The above equation is,
however, sufficient for the ocean as a whole. The individual terms will now be dis-
cussed in some detail.

(a) Direct Solar Radiation

Of the solar constant /^ is 1-94 g-cal cm~- min"^ one horizontal cm^ at the sur-
face of the Earth obtains for a zenith distance z of the sun (altitude /; = 90^ — r)
and due to the angle of incidence and the reduction in intensity due to the atmos-
pheric absorption only the intensity

/ = /q e~^" sec z cos r.
Sec z is the relative thickness of the air through which the radiation passes (equals 1
for an atmosphere pressure of 760 mm Hg with the sun at zenith; equal to 2 when
the sun is at an altitude of 30°). e-^« = ^ is the transmission coefficient and q has
under normal conditions a value between 0-6 and 0-7, a = 0-1 28-0-054 log sec z and
ris the "turbidity factor". If z and Tare known then the direct solar radiation inci-
dent per cm^ on a horizontal surface can be calculated directly for any altitude of the
sun.

Part of the solar energy reaching the sea surface will be reflected there. This part
depends on the angle of incidence, that means from the zenith distance of the sun.
Schmidt (1915) has calculated that due to the compensatory effects of solar radiation,
which decreases with increasing zenith distance and of the simultaneously increasing
reflection the intensity of the reflected radiation for approximate calculations can be
put as ^ = O-OlO-O-013/o. By using the known total amounts of heat received by
1 cm^ of the Earth's surface by direct solar radiation with an average value of the
transmission coefficient, and by knowing the reflection loss at the sea surface, it is
possible to calculate the amount of energy obtained by 1 cm^ of the sea surface in
one day. Table 37 gives the mean daily total sum for a year for q = 0-6-0-7. The
figures show that even assuming a continuously clear sky the equator receives barely
one-half and the pole only a fifth of the solar radiation incident on the upper atmos-
phere. When the transmission coefficient is 0-6 the entire surface of the Earth receives
only 44% of the theoretical amount of heat. This value will be still further reduced
by the presence of clouds. If the cloudiness is w (as a fraction of the visible sky) then
the radiation actually reaching the surface of the sea is only

^2 = (I " u') S,,

where S^ is the value given in Table 37.

Table 37. Mean total daily sums of direct solar radiation on
g cal/cm^ day^). (J^ = 1 -94 g cal/cm^ mir

a free

water surface

Latitude 0° 10° 20°

30°

40° 50°

60°

70°

80°

90°

«7 = 0-6 i 402 392 365

q = 0-7 493 481 452

322
402

270 211

341 274

155
206

105
146

109

60
94

The Three-dimensional Temperature Distribution and its Variation in Time 91

Since the radiation on the surface of the ocean is difficult to measure and only few
determinations have been made, Mosby (1936) has given an empirical equation for the
mean monthly and annual values of the radiation incident per cm^ on a horizontal
surface for given values of the mean altitude of the sun and of the mean cloudiness

Qs == kh{\ - 0-07 liv); g cal cm-^ min-^.

The bars indicate mean values and k is a factor which depends on the turbidity of the
atmosphere; at the equator it is 0-023, at 40^" latitude 0-024 and at 70° latitude 0-027.

{h) Diffuse Sky Radiation

During the day the surface of the Earth also receives general scattered short-wave
radiation from the atmosphere and also direct solar radiation reflected from clouds.
Estimates based on the direct measurement of total radiation (direct + diffuse radia-
tion) show that in general the average value of the diffuse sky radiation for the whole
Earth and for a cloudless sky amounts to about 56% of the total radiation on the
upper limit of the atmosphere. If we take this value as an average for all latitudes, for
a cloudiness u-, the direct radiation S2 will be increased by diffuse radiation amounting
to 0-56vr • Si. At the surface of the water this more or less generally scattered radiation
will suffer a reflection loss of 6-6%. The fraction of diffuse radiation from the sky
entering the water is thus given by D = 0-52h' • S^.

The effective back-radiation R^ is the difference between the radiation according to
the Stefan-Bo I tzmatm law (E = err*) and the long-wave radiation of the atmosphere
and depends, for a cloudless sky, on the absolute temperature T of the lowest layer of
the atmosphere and on the water-vapour pressure in this layer (e in mm Hg) (Ang-
strom, 1936). The effect of clouds is shown in a reduction of the effective back-radia-
tion and can be calculated if the cloudiness is given. With this equation it is possible
to calculate numerically the longwave radiation of the atmosphere for a given tem-
perature, water-vapour pressure and mean cloudiness. The effective back-radiation
can be measured directly, but such measurements have only seldom been made over
the sea. Angstrom has derived an empirical formula which has been given by Moller
in the following form

^eff = oT^[l - (0-210 + 0-174 X 10-»-»55eo)(l - 0-675vv)],

where a is the Stefan-Boltzmann radiation constant, T is the absolute temperature,
Re^ is the vapour pressure above the surface of the sea and w — as before — is the mean
cloudiness.

For the surface of the sea it can be rearranged to give

Q^ = 0-954ar4 - (7r[(0-210 + 0-174 x lO-oo^^e^^^i _ o-765m-)].

Since, as shown by Lauscher (1944), the radiation from a plane water surface is
decreased by 6-6% by back-reflection (see p. 60). The effective radiation is the first
loss in the heat balance (see Table 36, item 1 (heat loss)).

92 The Three-dimensional Temperature Distribution and its Variation in Time

(d) Evaporation (see Chapter VII).

A further debit item is the heat lost by evaporation. The amount of heat involved
can be easily found from the mean zonal values for evaporation (WiJST, 1922), since
for the evaporation of 1 mm of water from 1 cm^ of a water surface 60-65 g cal
are needed.

(<?) Convection (heat exchange between the ocean and the atmosphere)

Little is known of the transfer of heat from water to air by convection. From the
approximate calculations of Angstrom (1920) it can be concluded that for a difference
in temperature of TC between the water and the air (air temperature measured
60 cm above the surface) the mean heat transfer by convection is between 0-01 and
0-03 g cal/cm-2 min-^ For the mean temperature difference between the water
surface and the air which has been measured more accurately for the Atlantic Ocean
(KuHLBRODT, 1938 a, b) the convectional flux amounts on the average to about
0-014 gcal/cm-2 min"^ or the average heat loss from the surface of the sea results
to about 20 g cal/cm-2 per day. In warmer climates this value will be increased up to
about 0-030 g cal cm'^ min-^ which is about 45-50 g cal/cm^^ per day. These values
are only rough estimates of this heat loss which is too large to be neglected in the heat
budget of the ocean.

The heat transport by convection follows from the equation

Qh = -CpA (^ + r

where Cp is the specific heat of the air at constant pressure, A the turbulent exchange
coefficient (eddy conductivity), —ddjdz is the vertical temperature gradient of the air
above the water (positive since the temperature decreases with height) and y is the
adiabatic lapse rate. CpA replaces the thermal conductivity coefficient (see p. 50);
y can be neglected in the above equation since it is much smaller than ddjdz. For
stationary conditions, that is with constant heat flux through a horizontal unit sur-
face, the temperature changes rapidly with height near to the sea surface and there A
is very small. For larger distances A increases and the temperature decreases so that
A(dOldz) can remain constant.

If the surface of the sea is warmer than the air above, the air is heated from below.
The vertical stratification of the air is then unstable and as the air turbulence increases
the vertical heat transport becomes large. If the vertical temperature differences are
large this can lead to intensive atmospheric disturbances. On the other hand, heat is
transported from the atmosphere to the sea when the water is colder than the air above,
but the heat transferred by this process is not very large since it stabilizes the air.
The exchange A is then small and if the vertical stability is sufficiently large, turbulence
of the air and the corresponding downward heat flux then ceases.

If mean values for the heat gain and loss, described in the above discussion, are
calculated for different latitudes, a heat budget for the ocean surface can be drawn
up as shown in Table 38.

The Three-dimensional Temperature Distribution and its Variation in Time 93
Table 38. Heat budget of the total ocean (g cal/cm^ day^)

Latitude

10"

20 "=

30^ 40=

50° j 60"

70=

Heat gain

80° I 90=

Direct solar radiation after

allowing for cloudiness
Diffuse radiation

Total heat gain

202

255

267

233

171

107

166

129

106

368

384

373

332

269

202

153

112

39
36

Heat loss

Effective back-radiation
Evaporation heat
Convection

Total heat loss

143

160

338

133

116

121

126

125

278

214

177

159

-9

-12

-24

-47

131 137

6 0

20 20

157 i 157

Gains-losses

-72 -82

In this heat budget it has been tacitly assumed that the heat exchange through the
ocean surface occurs independently for each separate latitude belt. Therefore no meri-
dional heat exchange (by ocean currents and by horizontal mixing) was allowed to
occur.

The differences between heat gain and heat loss show that for lower latitudes north-
ward, until about 25° N. the gain in solar energy is greater than the loss, while between
about 45° latitude and the poles the back-radiation is dominating because only a
small consumption of heat occurs due to evaporation and convection. The excess of
the large tropical and subtropical area is, however, roughly equalled by the deficiency
of the higher latitudes so that, when the effect of meridional heat transport is taken
into account, it can be seen that with reasonable accuracy there is a heat equilibrium
for the entire ocean.

This meridional heat transport is largely due to the turbulent motion in the ocean
currents through lateral mixing (in meridional direction) (see Chap. Ill, 2e). If the eddy
coefficient of lateral mixing is denoted hy Ay (g cm~^ sec"^), the meridional temperature
gradient by ddjdy, then the heat W carried towards the north through a unit vertical
area is given by the equation

W = -c„A

d&
dy

The amount of heat transferred from south to north across latitude 25° is given in
Table 38 and it can be calculated from these values that for turbulence effective down
to a depth of 1000 m an amount W of heat, which is approx. 1 g cal cm-^ sec~S
will be transferred through a vertical area of 1 cm^. The mean horizontal tempera-
ture gradient at 25° latitude is about — 4°C per 10° of latitude which is —3-6 x 10-»
deg/cm. The above equation thus gives Ay ~ 3 x lO'^gcm"^ sec~^ This calcula-
tion for Ay is naturally a very rough one, but it gives a value for the lateral eddy
coefficient which corresponds rather well to more accurate other determinations. It

94 The Three-dimensional Temperature Distribution and its Variation in Time

is, however, certain that lateral mixing in ocean currents is a factor of considerable
importance for the horizontal distribution of the heat in the ocean and thus plays an
important role in the heat budget.

2. Heat Transport in the Sea : Absorption, Conduction, Thermo-haline and Dynamic
Convection (Turbulence)

The previous section gave an outline of the average heat amounts reaching the upper-
most layer of the ocean ; the question of what happens to this energy shall now be
considered. First of all, it can be expected that the radiation energy absorbed will
manifest itself as a rise in temperature.

{a) Temperature Change Caused by the Absorption of Radiation

The almost complete absorption of the solar radiation (direct and diffuse), and also
of the long-wave radiation of the atmosphere in the uppermost layers of the sea, must
cause large daily and annual variations in temperature if this heat is not conducted
in some way to the deeper layers. As given in Table 37, middle latitudes receive about
300 g cal cm~- per day from direct and diffuse solar radiation. 120 g cal of it would
be required for the evaporation of about 2 mm of water so that there would remain
approximately 1 80 g cal for heating the water mass and for producing a daily tem-
perature cycle. Of this amount the uppermost layer of 10 cm thickness absorbs
about 81 g cal, according to Table 20, while the top meter absorbs 1 15 g cal per day.

Table 39.

Fore- After-

Night Total

noon

Heat gain by absorption
Heat loss by radiation

+61
-20

+20
-20

-41

+81
-81

Diurnal variation

+41

-41

Under stationary conditions this energy gain must be re-radiated during daytime by
the water. The partition between day (incident and back-radiation) and night (back-
radiation) for the 10 cm layer will be roughly as shown in Table 39. The rise in tem-
perature of the top 10 cm of water, during the forenoon until the temperature maxi-
mum, is caused by the absorption of 41 g cal, while the rise for the top meter (100
cm^ of water) is derived from the absorption of 57 gcal; these amounts correspond
to a temperature range of 4-1 ° and 0-57°C. Therefore, the diurnal temperature changes
in the surface layer of the sea (and in lakes) may remain very small and are much less
than that of the land and the air immediately above it. During the summer half of the
year the gain during the day is greater than the loss during the night and heat is
accumulated in the uppermost layer.

These considerations raise the question of a possible radiational equilibrium within
the uppermost layers of the sea; only in this way can there be an appreciable absorp-
tion of radiation. In layers that are not too thick, water is somewhat more transparent
for short wave than for long wave radiation (see p. 52). Since the absorption

The Three-dimensional Temperature Distribution and its Variation in Time 95

coefficient is different for different wavelengths, water cannot be considered as a grey
radiator (Emden, 1913). It is, however, only for grey radiation that the final state of the
radiation equilibrium is an isothermal state, in which every layer absorbs just as much
energy as it gives off so that the temperature remains constant. For these reasons an
isothermal top layer (thin homogeneous layer of uniform density) thus cannot be in
iationalrad equilibrium with the solar radiation (direct and diffuse) (see Defant, 1936).

{b) Thermal Conductivity

If there exists a vertical temperature gradient in the water, then heat will be trans-
ferred from warmer to colder locations by the process of ordinary heat conduction.
There is a constant tendency towards equalization of temperature differences, and
this heat transport disappears only when there is a fully isothermal state. The question
of interest here is the speed of this process. From theoretical physics it can be shown
that the change of temperature with time for a temperature gradient ddjdz is given
by the differential equation

dd _ _A_ 8^9
dt Cpp dz^ '

In case of a horizontal (along x-axis) movement (velocity u) in the water

dd d'd' 8^9'

where dd'fdt is the local change of temperature with time. For no horizontal motion
(u = 0) the equation for the thermal conductivity takes the form

bd _ A d^d

'Ft ^c^p d^'

The solution of this equation (see, for example, Riemann-Weber, 1910) for different
boundary conditions provides the answer to important questions concerning the
temperature distribution in the sea. It is, for example, of interest to know how fast a
temperature change at the surface travels downwards within the water mass by thermal
conductivity. The numerical evaluation of the corresponding solution gives for
different depths the time required by the disturbance to reduce magnitude to half of
its surface value (half value-time). For a thermal conductivity

a = Xlc^p = 1-309 Xl0-3cm2/sec
one obtains the following values (Table 40). Millions of years would be required
for a temperature change at the surface of the sea to reach the larger ocean
depths. These values show in the clearest possible way the unimportance of thermal
conductivity for oceanographic phenomena, since there are other processes which
give a much faster propagation of temperature changes down to the ocean interior.

Table 40. Downward progression of a sudden temperature change in the sea by thermal
conductivity {time needed to reach the half value of the surface disturbance)

Depth (m) | 1 j 10 I 50 100 500 1000 3000 9000

Time (years) i 27 i 665 2660 66,500 i mill. I 2i mill, j 9 mill.

96 The Three-climensionol Temperature Distribution and its Variation in Time

Only in the absence of any more rapid processes could the lower temperature of the
deep sea be taken, as was previously assumed, as evidence of a much lower former
temperature at the surface of the ocean (ice ages). Periodic changes of temperature
at the surface of the sea will be transmitted to the deeper layers and cause a periodic
change there also. The theory of conductivity shows that when the surface change
has the simple form

&0 = Oq cos y /

it will have at a depth r the form

^2 = Qq e""" cos

(y'-4

where a = ■\/{TrjaT) = ■\/{Cj,pttIXT) and Tis the period of the oscillation. The amplitude
of the change in temperature decreases according to the e-function and at the same time
there is a phase shift. For the diurnal variation in temperature the amplitude at the
surface is reduced to 1% at a depth of 28 cm and the extremes at this depth have
already been shifted by three-quarters of the period (18 h). The corresponding values
for the annual variation are 5 m with here also a phase change of three-quarters of the
period (268 days). The general effect of molecular thermal conductivity confines both
these periodic changes to the very uppermost layers of the sea.

A much more rapid process than molecular thermal conductivity is the vertical
displacement of small quanta of water which occurs when a small part of a water mass
is heavier than the water underneath it. To restore the disturbed equilibrium the heavier
water tends to sink and the lighter to rise. Associated with these forced vertical move-
ments of small water quanta there is also a transport of the characteristic properties
of sea-water in vertical direction which leads to an equalization of any vertical
differences in these properties which may be present (see p. 195).

This has a rather important effect on the state of the deep water layers. An
increase in the weight of small water particles at the surface may be caused either by
an increase in salinity due to evaporation or by the formation of ice or it may be due
to cooHng. If the temperature of a small water particle falls, its specific volume also
decreases as long as the salinity is greater than 24-7%o (see p. 46). In a volume of
water with a horizontal cross-section of 1 cm^ and a height of // cm the temperature
change AS- due to a removal of an amount of heat AQ\^ given by

Cj>h

If a layer of water of e mm thickness evaporates from the top of such a column of
water with 5'%o salinity then the increase in salinity when evenly distributed over the
column of water is given with sufficient accuracy by

AS, = ^ €.

If at the top of a similar column an ice layer of e cm thickness is formed with a salt

The Three-dimensional Temperature Distribution and its Variation in Time 97

content 5^, smaller than S, then, as a first approximation the corresponding increase
in salinity is given by

0-9g {S - S,)

If the ice contains no salt (Sg = 0) then

In these quantities AQ,AS, and ASe (heat loss, salinity increase by evaporation and by
ice formation) lies the primary cause of every thermo-haline convection. In lower lati-
tudes where there are only small variations in the temperature the heat loss is out-
weighed by the effect of evaporation ; in temperate latitudes the heat loss by radiation
is the decisive factor, while in polar regions, in addition to these processes, the increase
in salinity due to the formation of ice is also effective.

Only very small changes in the specific volume are needed to initiate convection in
the uppermost surface layer since the resistance to be overcome is not large, a hun-
dredth %o salinity or a hundredth degree centigrade is sufficient.

The range of effectiveness of convection depends entirely on the vertical density
distribution in the water mass in which it occurs. For a given surface disturbance it
can only extend down to that depth at which the displaced quantum of surface water
reaches, water having the same specific volume. If there is a rapid decrease in the spe-
cific volume, then the convection will cease in the upper layers ; this is liable to occur
particularly at the density transition layer (thermocline) which acts as a barrier layer
and confines the thermo-haline convection to the top layer of the sea (thin homo-
geneous layer of uniform density). On the other hand, a randomly initiated disturbance
of any size at the surface leads to convection which extends in a homogeneous water
mass down to the bottom. The range of effectiveness of convection is a maximum only
when the density disturbance of the sinking water quantum is retained while it sinks.
If, as is to be expected, it mixes with the surrounding water the disturbance will be
rapidly decreased and the depth of influence of convection will be correspondingly
less. The larger the density difference between the sinking water and its surroundings
the more rapidly the difference between them will be diminished and the greater the
reduction in the depth of the convection layer.

When the sea has a normal stable structure (tropics, subtropics and temperate
latitudes) the nocturnal convection before sunrise will extend to a depth of 10 or 20 m.
The seasonal convection processes, caused by prolonged cooling during the autumn
and the winter, will extend to greater depths, normally to about 300 m. The con-
vection is developed to its greatest extent in polar and subpolar latitudes, where it is
assisted by a very uniform temperature and salinity distribution. The question for the
primary cause initiating these major convection processes, which are of decisive
importance for the deep-sea circulation of the ocean, has been the subject of a con-
troversy that is still not without interest. The initiation and maintenance of the vertical
convection in higher latitudes could be due to the cooling of the upper layers by radia-
tion, or it could be due principally to contact with melting ice. Pettersson (1904)
supported the strong cooling effect of the ice that is so plentiful in these latitudes,
while Nansen (1912) favoured the direct cooling of the surface layer by outgoing

98 The Three-dimensional Temperature Distribution and its Variation in Time

radiation. This controversy was settled by important and interesting experiments in
the sense of Nansen's reasoning. He suggested that the winter convection in parts of
the Norwegian Sea and of the North Atlantic (south and south-east of Greenland and
in the Irminger Sea) could reach very great depths because of the almost uniform den-
sity structure of the sea, so that the autumn and winter cooling thus continued almost
to the bottom. This should therefore be the place where the uniform North Atlantic
Deep Water was formed. The observations of the winter cruises of the "Meteor"
in the Iceland-Greenland waters during 1929-35 have shown that these views of
Nansen were correct. The cause of this convection is certainly the radiation of the
surface layer during the late autumn and early winter.

In the North Polar Basin conditions are somewhat different. The very large rivers
of Asia and North America bring large amounts of fresh water into this basin and these
overlay the saline water that flows into the deeper layers from the Atlantic Ocean.
Any deep-reaching convection is scarcely possible here, cooling is limited to the sur-
face layer and is correspondingly stronger. The melting of ice in the spring and summer
sets up a barrier against the denser water masses in the deeper layers so that the sum-
mer heating does not penetrate far.

Characteristic examples of a convection that extends to great depths, and can be
attributed primarily to an increase in the salinity of the surface layer caused by strong
evaporation, are found in the Mediterranean Sea and in the Red Sea. The low precipi-
tation, the small amount of river water flowing in, and the high rate of evaporation
raise the salinity of the surface layers especially in the summer, though at this time only
a limited convection occurs, since the increase in density is largely offset by the effect
of the summer heating. However, in the autumn and winter a well-developed con-
vection is set up due to the lowering of the temperature of the surface water and
reaches to great depths because of the uniformity of the vertical structure of the deeper
layers.

The accurate mathematical treatment of thermo-haline convection processes is not
easy. It can be attempted in the following way (Defant, 1949). To begin one considers
two thin layers of thickness h-^ and h^, temperature ^i and d'z, salinity ^i and S^ and
density pi and p^. A disturbance introduced in the entire upper layer so that p^ = p^
will cause mixing of the two layers /zj and h^ by convection and the final result will
be the layer h^ + h^ of density p^. If the disturbance in the upper layer is assumed to be
due entirely to a reduction in the temperature of the upper layer by '&-y — Ad'i then the
final temperature at the end of convection results to

(^, - A{^,)h, + ^Jh h,

— h^^. — ^^--^:;^^-

when the mean temperature that would be obtained by simple mixing of the initial
water masses is given by

^'•'^ h, + h, '
The final salinity after ceasing of convection is given by

Syh^ + S^h^

^'''~ h, + h,
and corresponds to the salinity obtained on simple mixing.

The Three-dimensional Temperature Distribution and its Variation in Time 99

If the disturbance in the upper layer is due to an increase in the salinity by AS^,
then the final temperature and salinity are

h
^1,2 and 5i,2+i-^ ^S^-

A disturbance in the second layer can in the same way be passed on to a third and from
this to a fourth and so on while at the same time its intensity decreases continually.
If the disturbance in the layer /?! + //g is due to a temperature decrease of J )?i,2 then
progression of the convection to the third layer in an analogous way gives the tem-
perature and saUnity at the end of the convection process as

/7iJl^l + //l, 2^^1,2

^1.2,

^1,2,J

and iSi, 2, 3'

If the disturbance is due to an increase in the salinity ofASx,^ then the temperature and
salinity are

t^i.2,3 and ^1,2, 3 H r .

"1,2.3

Thesimplestwayofcalculatingzl'!^andJ5'is to use a [r5]-diagram (see Chap. VI). In
Fig. 46 the thin Unes are lines of equal density (isopycnals). The point A shows the values

20°

347o<

35%<

36%,

377o<

15'

10'

25'.5

^ 26
X 26-5

/; ^y

^"y^

A\,

X y

/ 2

7 /b /

27.'5 /

/ /

28
/

/ /

/ 28.'^

29 /

/ ^

Fig. 46. [r5]-diagram for the determination of degree of disturbance during the initiation of
convection processes in the sea.

1 00 The Three-dimensional Temperature Distribution and its Variation in Time

of 19' and S of the upper layer hy. The density p^ of the second layer //o corresponds to
the isopycnal that passes through the points B and C. Since the density of the upper
layer must be equal to the density of the lower layer, as must be the case at the end
of the mixing by convection, then either the temperature must decrease hy AB =
—A'd'i if the salinity is constant, or the salinity must increase by AC — ASi if the
temperature is constant. A convenient connection of the point A with a point D on
the ispycnal p2, between B and C, gives the value of the disturbance for the tem-
perature and the salinity if both are present at the same time. The determination of
magnitude of the disturbances from the [r5]-diagram in this way offers little
difficulty,

A simple schematic diagram gives a convenient representation of the results of
convective mixing. Fiaure 47 shows the normal vertical distributions of d and S and of

%^^<=^ At AS

Fig. 47. Change in the thermo-haline structure of the sea
produced by convection processes.

the specific volume a; they represent the conditions before the mixing of the upper
layers by a convection extending only to a depth h. If the convectional disturbance is
entirely due to a reduction in & (by radiation) then the state of the upper layer at the
end of the convection process is characterized by the broken straight line; if, on the
other hand, the convection disturbance is entirely due to an increase in salinity the
dotted straight line shows the final state. It can be seen that the convection levels out
any differences in the vertical gradient for the different factors.

(d) Dynamic Convection (forced vertical mixing)

While thermo-haline convection is produced by external sources of disturbance
and continues as long as these disturbances remain, dynamic convection depends on the
forced mixing of superimposed layers of water embedded in a turbulent current. The
disordered eddying flow of larger quanta of water within such a current causes a con-
tinuous mixing of the water mass in both vertical and horizontal directions. This mixing

The Three-dimensional Temperature Distribution and its Variation in Time 101

process affects not only the vertical distribution of velocity within the current, but
also plays a considerable role for the distribution of the properties of the water mass.
The importance of such a mixing process, due to turbulent flow in a water mass, was
realized much earlier in oceanography than in meteorology. Gehrke (1909, 1912)
was the first to show that the mixing of the water masses in an ocean current must
give rise to a vertical transfer of heat. He found that this vertical heat transfer is pro-
portional to the product of the specific heat and the vertical temperature gradient, so
that it corresponds to the ordinary equation for the molecular thermal conductivity,
but with a coefficient which is dependent on the intensity of mixing and is consider-
ably larger than the coefficient for molecular thermal conductivity. Gehrke termed
this a "coefficient of turbulent mixing"; it has the dimensions [cm^ sec-^]. Following
Gehrke, Jacobsen (1913, 1915, 1918), in particular, has dealt in detail with the
"apparent" thermal conductivity and with the "apparent" diffusion which are con-
nected with turbulent processes. He pointed out that for all the processes initiated
by the mixing of the properties of the water (temperature, salinity and the content
in sea-water of other dissolved and suspended materials and of organisms) the tur-
bulent mixing coefficient should be the same and should be dependent only on the
intensity of the turbulence in the current. Through the turbulence also the flow mo-
mentum (impulse of the current) is affected by the "mixing" process, i.e. a vertical
equalization that manifests itself in the turbulent (apparent) viscosity. Already
Jacobsen has put forward the view that in the transfer of the small quanta of water
from layer to layer within the turbulent flow produces an immediate and complete
equalization of the momentum; however, complete equalization of the properties of
the water does not necessarily follow. This would imply that the "intensity of mixing"
of the momentum (turbulent viscosity coeflricient) must always be larger than that of,
for example, the temperature or the salinity (apparent thermal conductivity coefficient,
apparent diffusion coefficient). These views of Jacobsen appear to be confirmed by the
quantitative determination of these coefficients.

Following these investigations which gave a deep insight into the nature and
efficiency of turbulent ffow, Schmidt (1917, 1917^, 1925) and Taylor (1915, 1918,
1922), at about the same time, carried out extensive work on turbulent flow and on the
phenomena connected with it, which has had a wide utility for the explanation of
several oceanographic phenomena. These started from the basic approach that due to
the random movement of individual small quanta of water in a turbulent flow there
is not only an equalization of the momentum in the direction of the largest velocity
gradient, but that every property can be transferred to an adjacent mass in the direc-
tion of its largest gradient. The simplest derivation of the most important and funda-
mental equation for the interchange of properties within a turbulent flow has been
given by Schmidt. Consider a horizontal unit area (1 cm^) in such a horizontal flow,
whereby the vertical direction z is counted positive upwards and negative downwards
of it (the zero point (z = 0) lies in the surface itself). Due to the turbulence of the
flow there will pass through this unit area a mass of water m^ upwards and a mass
ma downwards. Since, however, there is on the average a displacement of the water
only in a horizontal direction it follows that over a long period of time Hmy, = lima.
Every small quantum of water will, however, carry its properties with it during its
turbulent displacement. If one of these properties is designated by s (for instance the

102 The Three-dimensional Temperature Distribution and its Variation in Time
salinity) and j is a function of z only, then at the unit surface as a first approximation

s = s, + ^z,

where Sf is the value in the surface where z = 0. Every small particle of water passing
through the surface from below will take with it an amount w„ s^, while those from
above will carry an amount m^ s^. The final exchange flux S through the unit surface
upwards can be expressed as the difference

S = I^m^Su — ^tn^Sai

whereby the summation has to be taken for all the small particles moving upwards
and downwards through the surface. Now

ds J ds

Su = -^z + 3^ ^w and s^ = Sf + —Za,

where the values of Zy are all negative and the values of z^ are all positive. This gives

8s
S = {Sm^z^ — i:maZa) ^^ .

Considering the different signs of z, the quantity in brackets gives a negative sum
—Em I z I , where every small mass m of water moving through the surface is now
multiplied by the initial absolute distance | z | from the unit surface. This sum de-
pends only on the state of turbulence of the flow. Schmidt has called it the
"Austausch (exchange) coefficient" t]. It has the dimensions g cm~^ sec~^. The basic
equation for the exchange is thus

The most important exchange quantities involved in oceanographic turbulent trans-
fer processes are: heat-temperature, salt-salinity, gas amount-gas content, number of
organisms-organism content. The flow momentum-flow speed also follows this law
(see later).

It appears that the assumption that every small quantum of water starts from its
initial position with a property s corresponding to the mean vertical distribution at
that point does not entirely accord with the actual conditions. Only for the pair
flow momentum-velocity does there appear to be a complete and immediate equaliza-
tion of the velocity diff'erences. For all other properties a correction must be applied
to the above basic equation. Ertel (1942) has attempted to take these circumstances
into account, and obtained the equation

ds ds

S = -^(X - 2n) -= - A j^.

The Three-dimensional Temperature Distribution and its Variation in Time 103

Thereby, it was assumed that a small particle of water passing through the unit sur-
face is not immediately mixed completely with the surrounding water but is mixed in
the proportion 1 : n. For the velocity in a turbulent flow, n would be equal to zero
and the exchange coefficient A for the property s (eddy conductivity and eddy diff"us-
ivity) would then be less than the eddy viscosity coefficient -q. Determinations of A
and T] also verify this. Table 41 gives list of such determinations measured in currents
in different parts of the oceans.

It can be seen that -q is of the order of 100-200 or more while A is of the order of
5^0, on the average about 20 g cm^^ sec^^ The ratio -qjA is of the order 5-20.
Taking an average value of about 10, then Afrj = l-2n, n = 0-45, that means that the
small quanta of water in random movement are mixed with the surrounding water
only to the extent of about 45% of their mass and accordingly the temperature and
salinity, for example, tend towards the values of their surroundings at this rate. This
value is not unreasonable considering the difficulty of mixing water of different densi-
ties and the constant tendency for water masses of different densities to separate
again.

The exchange equation applied to the pair heat-temperature has the same form as
that for the molecular thermal conductivity (p. 50), except that the thermal conduc-
tivity coefficient a = {^lcj,p) is replaced by the quantity CpA (specific heat x exchange
coefficient). The exchange coefficient A is of course not constant and will vary from
layer to layer. Taking a mean value of about 20 g cm-^ sec~^, then since Cj, is
approximately equal to 1, Cj,A will be about 15,000 times greater than a. The molecular
thermal conductivity is thus of no importance compared with the eddy conductivity
(dynamical convection). The thermal conductivity equation for turbulent heat trans-
port is therefore

d^_A 8^&

Tt~'^ a?'

where A is assumed to be independent of the depth. If this is not the case the equation

8^ _l 8 / 8^
Tt^'p 8z y-^

Temperature changes at the surface will be transmitted much more rapidly by turbu-
lent thermal conductivity down to the deep-ocean layers. For the process of molecular
heat conductivity surface disturbances were shown to require a half-value time of
some miUions of years (see Table 40), however, for conductivity it would take only
some hundreds of years according to Table 42. Indeed, in the upper layers surface
changes will penetrate downwards by turbulent action remarkably rapidly; only a few
days are required to spread completely through the layer down to 50 m. Periodic
changes will of course reach deeper. For values for A of 20 and 100 gcm~^ sec~^
the amplitude of a diurnal variation will decrease to 1/100 of its value at the surface
in 34 and 75 m, respectively. For the annual variation the corresponding values are
644, 1440 m, respectively. This corresponds better with the values given by tem-
perature observations.

104 The Three-dimensional Temperature Distribution and its Variation in Time

{e) Horizontal Convection and Lateral Mixing

The dynamic convection discussed in the preceding section applies only to mixing
of water masses in a vertical direction moving in a horizontal turbulent flow. In
addition to this vertical mixing process there will also be a mixing process, largely

Table 41. Coefficients of eddy conductivity, eddy diffusivity and eddy viscosity

Coefficient

Current or oceanic
region

Depth of
layer (m)

Magnitude
(g cm~^ sec~^)

Reference

Eddy conductivity

Philippine Trench

5000-9788

20-3-2

Schmidt, 1917

and diffusivity

Algerian Coast

0- 20

35^0

Schmidt, 1917

from temperature

Mediterranean

0- 28

Schmidt, 1917

and salinity

Cahfornia Current

0- 200

30-40

McEwen, 1919

measurements, A

Caspian Sea

0- 100

1- 3

Stockman, 1936

Barents Sea

—

4-14

Subov, 1938

Bay of Biscay

0- 100

2-16

Fjeldstad, 1933

Equatorial Atlantic

Ocean

0- 50

320

Defant, 1932

Randesfjord

0- 15

01-0-4

Jacobsen, 1913

Schultz Grund

0- 25

004-0-74

Jacobsen, 1913

Kuroshio

0- 200

30-80

Sverdrup-Staff, 1942

Kuroshio

0- 400

7-90

Suda, 1936

Southern Atlantic

Ocean

400-1400

5-10

Defant, 1936

Arctic Ocean

200- 400

20-50

Sverdrup, 1933

Carribean Sea

500- 700

2-8

SeiweU, 1938

South Atlantic Ocean

3000-Bottom

Defant, 1936

South Atlantic Ocean

Near Bottom

Wattenberg, 1935

Eddy viscosity -q

Wind currents

Surface layer

l-OIw^w < 6
m/sec)

Thorade 1913/1914,
1914

Wind currents

Surface layer

4-3 (^^(m' > 6
m/sec)

Ekman, 1905

North Siberian Shelf

0-60 (tide)

75-260

Sverdrup, 1926

North Siberian Shelf

0-60 (tide)

10-400

Fjeldstad, 1936

North Siberian Shelf

0-22

H^m'

Fjeldstad, 1929

Schultz Grund

0-15

1 •9-3-8

Jacobsen, 1913

Caspian Sea

0-100

0-224

Stockman, 1936

Kuroshio

0-200

680-7500

Suda, 1936

Japan Sea

0-200

150-1460

Suda, 1936

Table 42. Advance of a sudden temperature change penetrating into the sea by thermal
turbulent conductivity {half value time of surface disturbance)

Depth (m)

1 10 50

100

500

1000

3000

6000

Time when Alp =

20 cm^/sec
Time when Ajp =

lOOcm^/sec

9 min 1 5 h 16 days 64 days
1-8 min 3h | 3 days 13 days

4-4 years
320 days

17-4 years
3 J years

185 years
37 years

624 years
125 years

The Three-dimensional Temperature Distribution and its Variation in Time 1 05

effective in the horizontal direction caused by currents moving side by side carrying
small masses of water at greater or lesser velocity and by eddies of varying size with
vertical axes, that is, by the lateral turbulence in the flow.

In the horizontal direction the disturbances are of greater dimensions than in the
vertical direction, particularly those due to atmospheric effects (wind, squalls and
rapid changes of pressure), which affect the surface layer of the sea and to some extent
the deepei layers also. Disturbances due to coastal and bottom topography are also
able to produce turbulence in an horizontal direction with turbulence elements which
must obviously develop on a much larger scale than the vertical turbulence. The
corresponding exchange coefficient will be much larger than for vertical mixing. In a
certain sense there is an analogy with the large-scale lateral turbulence in the atmos-
phere which is also quasi-horizontal (isentropic). In this case the coefficient is on the
average of the order of 10^ g cm~^ sec~^ as compared with an average value of 50-
100 of ordinary vertical turbulence. That lateral large-scale turbulence is also im-
portant in oceanic phenomena was first pointed out by Defant (1926), who determined
the order of magnitude of this exchange coefficient as about 5 x 10^. Later, Witting
(1933) discussed both vertical mixing and lateral mixing, and has attempted the de-
termination of the exchange coefficient by large-scale coloration experiments. Rossby
and co-workers (1936) have clearly shown that there occurs in the ocean, as in the
atmosphere, a lateral mixing of this type along the isotropic surfaces, which is essen-
tially in the ocean the same as along the or^-surface. Parr (1938) has shown the large
effect of this lateral mixing on the distribution of temperature and salinity in the water
masses around Newfoundland; Sverdrup and Fleming (1941) have found the same
effect in the coastal water off California and Stommel (1950) has determined the lateral
mixing coefficient Ajp in the Gulf Stream to be 2-3 x 10^ cm^/sec.

For a given horizontal gradient in any of the properties of a mass of water the
horizontal convection will play a large part in the long-period equalization of this
gradient. This presupposes a transport of the property along the direction of the gra-
dient. Furthermore, if a small mass of water has a property s (for instance, temperature)
present in amount S (for instance, heat), then the horizontal transport of S across the
horizontal turbulent flow in the direction n is as before, Sn = —An(Ssldn). An is now the
horizontal exchange coefficient. Its order of magnitude is several times larger than that
of the coefficient for vertical mixing A^. Since, in general, the vertical gradient of a
water property (such as temperature, sahnity) dsjdz is considerably larger than that in
the horizontal direction dsjdn, the horizontal transport Sn may still be of the same order
as the vertical transport S^, since in the above equation the product of the two quan-
tities is essential. This appears to be the case in reality so that lateral mixing is no less
important than the vertical.

Consider a volume element dx, dy, dz through which there is a turbulent flow with
velocity components u, v, w; the exchange coefficients in the three directions A a;, Ay,
Ay. Then, for the individual change with time in the property s the following equation
will apply

ds 8s 8s 8s 8s \ f 8 / 8s\ 8 / 8s\ 8 / 8s]

dI = 8t-^''8x-^'8y+''8-z--p[8x [^^8xj^8y l^^a^j + ^:^ l^^FrJ

If the .Y-axis is taken as the direction of the turbulent flow (positive in the flow

106 The Three-dimensional Temperature Distribution and its Variation in Time

direction) {v — w = 0), then stationary conditions in the distribution of the property
s in the volume element {^dsfdt = 0) are only possible if the equation

d'^s dh d^s 8s

is satisfied, where A;^, Ay and A^ are taken as constants. From this general equation
can be derived more special cases :

d^s ^•y « / X

^* aF^ - ^" a^ ^ ^ ^^^

if there is vertical mixing only (A^. = Ay = 0);

8^s ^s ^ • /,x

if there is transverse mixing (in a horizontal direction normal to the flow) (Ax = A:i =

0);

8^5 8^s ^'-^ _ n

if there is mixing in all directions but no aveiage water transport in the ^r-direction
(u = 0).

Cases (a) and (b) are mathematically identical but solely the vertical and horizontal
directions are interchanged. A solution for the equations (a) and (b) has been given by
Defant (1929)

77 TT^ Az

s = Sa + m e""* cos -^, z and a. —-r^ — .

For case (b), the co-ordinate z is replaced by the co-ordinate y. The distribution of the
property s along the homogeneous turbulent flow has been found to be tongue-shaped
if the 5-content initially has a maximum value at the centre of the flow (at x = 0,
s = Sq -{- m cos (7r/2/)z).

This is also the case when the velocity is the same over the whole transverse cross-
section. Figure 48 gives an example of the course of the i--lines for Aj p = 4 cm^sec,
M = 10 cm/sec and / = 2 x 10'* cm. The further the cross-section is taken from the
initial section {x = 0) the lesser are the horizontal and vertical differences in s. By
the extension of the above solution to different initial conditions for x = 0 (Thorade,
1931) it became evident that neither the tongue-form of the distribution of the proper-
ty s nor the distribution of the velocity u in the cross-section is considerably affected.
In addition, the initial distribution of 5 at x = 0 has equally little effect. The tongue-
form of the j'-curves is always re-established in a short time and is very largely a
consequence of the turbulent mixing. This is shown particularly well in Fig. 48a, which
shows the distribution of the property s in the case where the velocity is constant across
the transverse section, and initially for a: = 0 the property s is constant within the
distance 2/ {s = 100), while outside of this range there is no content of ^ in the water
{s = 0). In the flow a tongue-shaped distribution of s is produced immediately. This

The Three-dimensional Temperature Distribution and its Variation in Time 107

0 1000 2000 3000 WOO 5000 km

SOOOlm '

Fig. 48. Formation of a tongue-shaped distribution in a property of sea water by advection

and mixing (turbulence).

zooo

3000

iWGO

5000 km

5000 hm

Fig. 48a. Tongue form produced by turbulent mixing at constant flow velocity shown in a

cross-section (tongue of i'-content for a steady current, which attains a constant ^-content

in its total cross-section when it enters into a second water type.)

case corresponds to the conditions present in the spreading of a current of water of
high saHnity penetrating into a body of water of lower salinity.

In the horizontal and vertical distribution of the temperature and saHnity over a
large space in the ocean there are often found cases wheie the isolines have a tongue-
form. This distribution allows the numerical determination of the relationship be-
tween the exchange and the velocity of the flow, that is, of the quantity Ajpu provided
that this is imposed by exchange processes. Such calculations are fairly numerous:
they have been made, for instance, by Defant (1936) for the subantarctic intermediate
current and for the Antarctic bottom current in the South Atlantic (AJpu =-- 1 — 10
which for u = 1-5 cm/sec gives A^ as about 0-5-10 gcm-\sec-^); by Montgomery
(1939) for the equatorial counter current in the Atlantic (maximum value for A^
0-4 g cm-Vsec-\ for Ay 4 x 10^ g cm-Vsec-^ by Sverdrup and Fleming (1941)
for the coastal water off California at 200 and 400 m depths {AJp = 2x10*' cm^/sec)
and by Seiwell for the distribution of temperature and salinity in the Caribbean
Sea {AJp larger than 1C« cm^/sec). Recently Defant (1955) in an investigation of the

] 08 The Three-dimensional Temperature Distribution and its Variation in Time

spreading of the Mediterranean water into the North Atlantic found a horizontal
exchange coefficient of 5-5 x 10^ cm^/sec.

There is no doubt that the exchange coefficients for lateral mixing A^ and Ay are
about a million times larger than that for vertical mixing. The lateral mixing has thus,
despite the low values of the horizontal gradients for the different properties of sea-
water, at least the same importance as the vertical exchange. It can, however, be stressed
that the nature and inner mechanism of these two exchange processes are different;
the vertical mixing is small-scale, the lateral operates over a large space. It may be
expected that they are related to different ranges in the total turbulence spectrum
(see Chap. XIII, 3).

The third special case is for mixing operating in all directions but without any dis-
placement of water in a particular direction ; it shows therefore the effect of mixing
alone unaffected by advection. In the two-dimensional case (.v- and r-directions) the
solution takes the form (Sverdrup, 1940)

s = Sq + m

cosh [a{h — z)]
cosh [ah]

sin 27-v,

whereby

4/2

For z = 0, that is at the surface of the sea, the distribution of a property s is
s = Sq -{- m sm {ttI21)x.

Selecting, for example, h = 4 km, Sq = 0, m = 5 and AJA^ = 6 x 10^, then
a = 0-384 and Fig. 49 gives the distribution of s in an ocean of a horizontal extent

Fig. 49. Distribution of a property "5" in the total ocean due to mixing alone (according

to Sverdrup).

2/ = 20,000 km. In this case it would reach from pole to pole. The abscissa in Fig.
49 is therefore divided into meridional degrees from 90° N. to 90° S. For a per-
sistent maximum accumulation of the property s at the surface of the sea in equa-
torial regions, the effect of mixing alone would in the stationary state force a distribu-
tion of j: in ocean space shown by curves of equal s in this representation. For a per-
sistent temperature difference at the sea surface along a meridian, essentially the same
as that produced by the combined effect of the solar and back-radiation, the effect of
a mixing process acting alone ovei the entire ocean would give a vertical temperature
distribution such as that shown by the isotherms in Fig. 49. The temperature decreases

The Three-dimensional Temperature Distribution and its Variation in Time 109

everywhere with increasing depth, most rapidly at the equator and least at the poles.
This case will be considered later in connection with the actual temperature distribu-
tion in the deep ocean (see p. 123).

Another solution for the third special case is

^• = ^11

m e'

cos

z with jS =

11 ^ 2/

Obviously the solution of this distribution of 5 is identical with that of the first case on
p. 106 if j8 is put equal to a, that is if

This means that in a vertical cross-section a tongue-shaped distribution of s can be
equally well regarded as the effect of a horizontal advection with velocity u in the
direction of the tongue and as a vertical turbulence with an exchange coefficient A'^,
or as the sole effect of pure mixing in horizontal and vertical directions without any
advection. See Vol. I, Part II, Chap. XIII, 3, for a theoretical discussion of turbu-
lent mixing in ocean currents.

3. Diurnal and Annual Variation of the Temperature in the Ocean

The daily variations in temperature at the surface of large bodies of water (lakes
and seas) are confined within narrow limits as was mentioned previously. In lakes,
away from the shore, there may be diurnal variations exceeding 2 °C. They decrease
rapidly with depth so that at 4-6 m they may be not more than 0-1 °C (see particu-
larly the investigations by Homen (1913) in Lake Logo (Finland). Some idea of
the diurnal temperature variation (of the air and the water) is afforded by the in-
vestigation of Merz (1911) in the Gulf of Trieste (an enlcosed basin, relatively close
to the land). The amplitude of the water temperature was 0-87 °C, for the air it was
3-l°C, which is considerably more. For a discussion of the diurnal and annual varia-
tions of the surface temperature in a shallow water especially in the North Sea and
in the Bahic, see Dietrich (1953).

(a) The Diurnal Temperature Variation in the Open Sea

The diurnal variations of temperature in the open sea are even smaller than in
lakes; usually smaller than 0-4 °C and can rise at the most to about 1 °C in calm and
fair weather. The most accurate measurements of the daily temperature variation in
the open sea are obtained at anchor stations (fixed location). Four equatorial stations
between 12-5° N. and 4° S. of the "Meteor" Expedition in the Atlantic Ocean (De-
FANT, 1932) gave the following values (Table 43).

Table 43. Mean daily temperature variation from four "'Meteor'' anchor stations

Local

Ampli-

time

22 tude

(hours)

dt (°C)

-7

-10

-12*

-10

-8

-1

+ 11

+ 19t

+ 15

-1

-5 1 31

Minimum; j Maximum

1 10 The Three-dimensional Temperature Distribution and its Variation in Time

Latitude

12i°N.

4° N.^° S.

8°-14° S.

211° S.

Mean

Diurnal variation
At the surface (°C)
At 50 m depth (°C)

019
0034

0-40
0056

0-23
<005

016
(009)

0-25
(004)

The diurnal temperature variations always decrease towards higher latitudes; the
maximum occurs at 14.00 h and the minimum at 04.00-05.00 h. The diurnal course
corresponds almost exactly to a pure sine curve. KuHLBRorx (1938) obtained the same
results from a study of the daily temperature records of the "Meteor" Expedition by
the elimination of the effect of changes in position of the vessel. The average daily
amplitude for all areas of the South Atlantic amounts to only 0-26 °C. This value,
which was obtained by averaging all days without any selection, is somewhat smaller
than the value obtained by Wegemann (1920) from the "Challenger" observations
and by Meinardus (1929) from the "Gauss" observations.

The heating of the sea surface begins soon after sunrise due to the absorption of
solar radiation in the uppermost layer of the water, but the largest part of the added
heat is used for the evaporation of water (about two-thirds) and only a small part
remains for a temperature rise. The temperature thus rises only slowly to the maximum
at 14.00 h. After sunset the temperature fall continues due to outgoing radiation.

There are very few measurements of the depth to which the diurnal temperature
variation penetrates. The only information for 50 m depth is given by the hourly
observations at the anchor stations. However, for these depths near the thermocline
the influence of tides through the associated vertical currents (internal tide waves)
cannot be entirely excluded. Table 43 contains some values for the diurnal temperature
variation at 50 m depth showing that for these depths the amplitude is less than
0-05 °C. Aime has made measurements of the diurnal temperature variation at different
depths off the Algerian coast. It is, however, not entirely certain that all the observed
changes can be attributed to the diurnal cycle; however, if this assumption is made it is
found that the nocturnal cooling at 14 m is one-fifth of the surface amplitude and that
the heating during the day, which is three to four times stronger than the nocturnal
cooling, falls to a tenth at 28 m. Schmidt (1925) calculated from this decrease of
temperature the vertical exchange coefficient as 35-40 g cm"^ sec~^. The observations
of Knott on the "Pola" Expedition in the eastern Mediterranean show a decrease in
the amplitude to a tenth at 29 m, which corresponds to an exchange coefficient of
42 g cm~^ sec~^. Since the ocean covers more than two-thirds of the surface of the
Earth it can be said that over much the largest part of the surface the diurnal tem-
perature variations remains less than half a degree. Therefore, the considerably
greater diurnal temperature variations of the continents play only a minor part in the
total heat budget for the Earth.

{b) The Annual Temperature Variation

Changes in temperature over longer periods can be investigated in two different
ways. They can be recorded as "individual" temperature changes in a water mass
which is followed in its course in the ocean; they are then described by reference to
"oceanographic" co-ordinates. On the other hand, they can be followed at fixed

The Three-dimensional Temperature Distribution and its Variation in Time 1 1 1

points and are then referred to "geographic" co-ordinates. These last changes are
more complicated, since they are a combination of thermal changes within an indi-
vidual water mass and of changes caused by the displacement of different water bodies
(ocean currents). t

For most parts of the ocean the annual displacements of the currents are known and
most of the major annual changes in these areas can be ascribed to these. The seasonal
displacements of the Gulf Stream system and the Labrador Current in the region of
the Grand Banks of Newfoundland are well known. The large annual temperature
variations in this region of the sea are associated with these displacements Similar
conditions are found off the Norwegian coast where the seasonal displacements of the
coastal current and the Atlantic current cause pronounced seasonal variations in
temperature and salinity.

For smaller areas the annual temperature variation at the sea surface can be derived
only from a statistical evaluation of ship's observations and for the deeper layers from
series observations made by oceanographic expeditions. Averaging the values that fall
for different parts of the year into one, two or more degree squares gives mean tem-
peratures for these subsections of the year with sufficient accuracy, provided there is a
reasonable number of observations available. This of course gives only values related
to "geographic" co-ordinates. Such a rough statistical method can only be used with
some reliability for the sea surface.

All the available data on surface temperature in the Atlantic Ocean have been
collected and studied by Bohnecke (1936) and presented in a comprehensive form.
For the Indian and Pacific Oceans a less complete presentation has been given by
SCHOTT (1942). These show that there is an absolute minimum in the annual variation
of surface temperature of all oceans in the tropics where over extended areas, especially
in the Indian and Pacific Oceans, this variation is less than 1 °C. There is also a second-
ary minimum in the Southern Hemisphere everywhere in the water encircling the
Antarctic continent which also shows values less than 1 °C. In the Northern Hemi-
sphere there is a decrease in the annual temperature variation in the Norwegian Sea
and the variation becomes gradually smaller towards the north; this is true also for the
North Pacific, but the northward decrease is slower. The maximum annual tempera-
ture variation always occurs in the subtropical high-pressure belt where, near to the
Bermudas and near the Azores, the maximum value is greater than 8°C. This region
is connected with that showing the absolute maximum surface temperature variation

t The individual change in temperature d?^ldt in a given unit mass is caused by the addition or
abstraction of a given quantity of heat Q. This quantity Q is due to the absorption of radiation, to
back-radiation, to thermal conductivity, to evaporation and to mixing and others. If the local distri-
bution and that with time of these properties is given along the path followed by the unit mass of
water then the "individual" variation in temperature d^ldt can be found. If the "local" temperature
change d^jdt is required for a fixed point occupied successively by different masses of water, then for
a given flow (velocity u) in the direction n the following equation is valid:

db db db 1

dt ct dn Cp

The advection term u(8bl8x), which includes the effects of the transport and the displacement of
different masses of water at different temperatures in the direction n, thus plays an important role
for the assessment of the local temperature change d^ldt.

1 1 2 The Three-dimensional Temperature Distribution and its Variation in Time

(larger than 15 °C) off the coast of North America and in the region of the Newfound-
land Banks. There, as already mentioned, the annual variation in temperature is
caused by the fluctuating seasonal movements of ocean currents. Similar conditions
occur in the North Pacific; the absolute maxima of more than 20 °C in the Yellow
Sea, and in the Sea of Japan, are associated with a zone of maximum annual ampli-
tude (greater than 9°C) extending from Japan eastward towards the east coast of
North America. In the Southern Hemisphere the subtropical maxima of temperature
variations are of a smaller extension. The annual temperature range is also large
(8-10°C) in the areas of cold water upwelling (off" western Africa in the Northern and
Southern Hemispheres and off California) in accordance with the seasonal variations
in these phenomena.

The geographical distribution of the annual temperature variation at the sea
surface is not difficult to explain. In the tropics the small amplitude is due in the first
place to the constant high altitude of the sun throughout the whole year and also to
the relatively high cloudiness, so that there are only small annual variations in the in-
coming radiation. In the subtropics the absorption of solar radiation has a much
greater influence on the development of a marked annual temperature variation be-
cause of the already larger seasonal changes in the zenith distance of the sun, and also
because of the stronger effect of back-radiation due to the low cloudiness prevalent in
these areas. With increasing latitude the incoming radiation becomes less effective
and the autumn and winter convection, which is able to penetrate down to greater
depths here, still further reduces the annual amplitude of the temperature variation
until it reaches a minimum in the polar regions. Table 44 shows mean annual variations
in temperature for equatorial, temperate and high latitudes, by the use of the mean
temperatures for zones of 10° latitude given by Bohnecke (1938). In the equatorial
zone there are two maxima at the time of the equinoxes. In the subtropics the maximum
occurs in September and March, respectively, and in the extra-tropical regions in
August and February, respectively. The minimum values in the first area occur in
March and August respectively, and in the latter, in February and September, re-
spectively.

Table 44. Annual variation in the water temperature at the sea surface in the

Atlantic
(Deviation from annual mean 0-1 °C)

_ c
a 0

Latitude

Jan.

Feb. Mar.

Apr. May

June

July

Aug.

Sept. ! Oct.

Nov.

Dec.

50 '^-70" N.

-16

-21*

-13

-17 -11

+ 15

+26t

+ 13

+ 6

-5

- 6

4-7°

10°-50° N.

-23

-29

-30*

-24

-12

+28

+38t

+34

+ 19

-14

6-8°

20=N.-20°S.

+ 1-5

+2-5

+ 7

+8-5t

+ 6

- 5

- 8*

- 6

- 2

-0-5

+ 1

1-7°

20°-50° S.

+ 19

+27t

+23

+ 15

+ 4

-8

-15

-20

-23*

-18

-8

+ 7

50"

70^N.-60 S.

- 1

- 3

- 5*

-1 : 0 +4

+ 2

-1

+ 1

0-9°

* Minimum; f Maximum

In general the surface temperature minimum is retarded about two to three months
after the sun reaches its lowest height; the maximum is retarded also by about the

The Three-dimensional Temperature Distribution and its Variation in Time 1 1 3

same. A comparison of the oceanic and continental annual temperature variations is
given in Table 45.

Table 45. Annual temperature variations (°C)

Latitude

Equator

10°

20°

30 =

40°

50^

Oceans
Continents

2-3
(1-3)

2-4
3-3

3-6

7-2

5-9
10-2

7-5
140

5-6

24-4*

* Only Northern Hemisphere

Figure 50 shows isopleths for the annual surface temperature variations in the At-
lantic. It can be seen at once, that there is a narrow zone just north of the equator,
where there is a six-monthly temperature variation so that over the whole of the tropics
the amplitude of the annual temperature variation remains very small and that the
middle latitudes between 30° and 50° show a maximum which decreases towards the
pole, especially in the Southern Hemisphere.

The annual temperature variation is transmitted to the deeper layers beneath the
surface by the effect of convection and turbulence, with a corresponding reduction in
amplitude and a retardation of the extremes, until it finally disappears. However,
the annual displacements of water masses can also simulate an annual temperature
variation, which is then not due to the total production and expenditure rates of heat
at that point, but to others at more distant parts of the sea. Our present knowledge
of these phenomena is still very poor. To obtain the exact annual temperature varia-
tion at deeper layers it is necessary, because of the small number of observations

Jan. Feb. Mor. April Moy June July Aug Sept. Oct. Nov. Dec. Joa

Fig. 50. Isopleths of surface temperature in the Atlantic Ocean.

114 The Three-dimensional Temperature Distribution and its Variation in Time

available, to eliminate aperiodic changes. This elimination is done by the "tempera-
ture anomaly" method given by Helland-Hansen (1930). According to the mean
[rS'l-diagram (see Chap. VI, 3) there is normally, for every value of the salinity in the
water mass under consideration, a definite mean temperature {>. If an observation
(^1, S) is obtained in this area, then the difference ?^i — i^ is termed the "temperature
anomaly" of this observation. Experience shows that the temperature anomalies in a
given set of data is of a considerably smaller scatter than the original values and that
the aperiodic change of it has very largely been eliminated. The annual temperature
variation in particular is shown much better than by the original values.

By these methods Helland-Hansen has worked out the annual temperature variation
for the water layer down to 200 m depth for three ocean areas in the North Atlantic.
Figure 51 gives the results for the Bay of Biscay (area B) for the surface, as well as for

211

vnr

'^-

/ f

/ /

' /

,--

C'.

--.r~-

,rf5<

■z^„-^

' —

■ — 1

N-

Fig. 51. Annual temperature variation in the water layer down to 100 m depth in the Bay
of Biscay (area B) (according to Helland-Hansen).

the depths of 25, 50 and 100 m. Table 46 presents the time of occurrence of the
maxima and minima in this area and in the area between Portugal, Morocco and
Madeira and also gives the amplitude at different depths. The amplitude at 25 m
is still quite considerable and not very much smaller than the surface amplitude.
However, lower down it decreases more rapidly and at 200 m the annual variation is
more or less insignificant. The shape of the curve is almost the same in both areas and
quite characteristic. In late autumn and in winter the surface water cools rapidly and
the resulting convection also involves the deeper layers in this coohng. Thus, the ver-
tical temperature gradient decreases continuously and becomes almost zero in spring.
Heating now raises rather rapidly the temperature of the uppermost 25 m layer.

Table 46. Annual temperature variation in the Bay of Biscay (B)
and in the area between Portugal, Morocco and Madeira (C)

AreaB

AreaC

Depth

Min.

Max.

Variation

Min.

Max.

Variation

(m)

(°C)

Jan.

Aug.

7-7

Feb.

Sept.

5-3

Feb.

Aug.-Sept.

6-8

Feb.

Sept.

4-7

Mar.

Sept.-Oct.

2-4

Mar.

Oct.

1-4

100

Mar.

Dec.

0-7

Mar.-Apr.

Nov.

0-9

200

—

0-3

—

0-25

The Three-dimensional Temperature Distribution and its Variation in Time 115

This effect may still be noted down to 50 or 100 m, but the heating process is inter-
rupted in June and only reappears later at 50 m. The reason for this remarkable
phenomena can be seen in the circumstance that turbulence due to the wind affects
the greater depths in spring, while the water mass processes an indifferent or weakly
stable stratification, so that surface heat can penetrate still to depths below 50 m.
The rapid temperature rise of the surface layers soon builds up such a strong tempera-
ture gradient thai turbulence is unable to prove a match for the created strong vertical
stability of the water masses and the turbulent transport of heat therefore ceases. The
upper layer is heated further by continued incoming radiation, and because of mixing
becomes almost isothermal while the lower layers remain cold. The temperature in
these layers rises again. Only when the density gradient is destroyed in the autumn
can the effect of mixing and convection again extend to deeper layers. Only then can
further heat be carried to the layer beneath the thermocline (Defant, 1936fl).

In places where the ocean currents are subjected to considerable displacements, in
both direction and strength during the year, the annual temperature variation can
be considerably affected down to great depths by these current displacements. A
typical example for this is the annual temperature variation in Monterey Bay, Cali-
fornia (Skogsberg, 1936). Here there are three different periods in the annual variation:
the period of the Davidson Current from the middle of November to the middle of
February, when the temperature varies only slightly with depth down to almost 100 m;
then follows a period of upwelling water from the middle of February to the end of
July with low temperatures and stronger stratification ; while from the middle of July
to the middle of November the Californian Current prevails and the temperature
variation shows normal oceanic conditions.

On the other hand, the temperature variation in the Kuroshio south of Japan
(KoENUMA, 1939) shows almost exactly the same conditions as in the Bay of Biscay
which was mentioned above.

Fjeldstad (1933) has attempted to use the observations of Helland-Hansen in
area B to calculate the eddy conductivity coefficient from the changes in the annual
temperatuie variation with depth. He developed the annual temperature variation in
the individual depths into harmonic series, and obtained in that way the values c„
and a„ as the amplitude and phase of the «th term of the series. Fjeldstad then showed
that

A na

dz.

p K (^««/^^)

where a = l-n-jT, T is the annual period and h is the depth at which the ampUtude
vanishes. A better representation of the observations can only be achieved by assum-
ing a seasonal variation of the eddy conductivity coefficient. The mean value at the
surface is 16gcm-isec-\ at 25 m it is 3gcm-^sec-^ and at 100 m the annual
mean is only 3, in summer 0-5 and in winter 5-5.

The same method has been applied by Sverdrup (1940) to values for the Kuroshio
which in this case appears to be permissible, since the advective effects are outweighed
by radiation and the eddy conductivity. In the Kuroshio area, where the strength of
the current is large and the turbulence correspondingly high, the annual temperature

1 1 6 The Three-dimensional Temperature Distribution and its Variation in Time

variation penetrates almost to 300 m depth with an eddy coefficient of about
70 g cm~^ sec"^ at the surface, 30 and 27 at 50 m, and 200 m, respectively.

For a selection of 1 and 2 degree squares in the area between the Faroes and the
Bay of Biscay, Neumann (1940, unpublished manuscript) using simple statistical
methods has derived the annual temperature variation for 20, 40 and 100 m and has
obtained rather similar results. The effect of stratification on heat transport in the
deeper layers of a water mass also appears in lakes in the same way as described above
and the annual temperature variation can be explained only under consideration of
these processes. In shallow seas (Shelf seas) it is possible to study more accurately
the penetration of heat and especially the eff'ect of turbulence generated by the wind,
and also to investigate the eddy viscosity caused by strong tidal currents at the sea
bottom. Recently Dietrich (1950, 1953, 1954) has given an instructive example of
the various possibilities by the use of isopleths of temperature in vertical cross-sections
in diff'erent shelf waters which illustrates the conditions present in the best possible
way. Figure 5\a shows the annual variation of temperature and salinity from the sur-
face to the bottom.

31-1 35-0 35-1

Fig. 5\a. Example of annual temperature and salinity variations from the surface down to
the bottom (according to Dietrich), a, Irish Sea, North-Channel; b. Central North Sea;

c, Baltic, Bomholm deep.

{a) In the Irish Sea, north channel, with strong tidal currents where even in summer a
thermocline cannot form and the strong turbulence evens out the annual temperature
variation in the whole mass of water from the surface to the bottom (extremely strong
heat transport from the surface downwards due to turbulence).

{b) In the middle of the North Sea where the weak tidal current has little eff'ect.
The formation of a thermocline prevents the development of a more pronounced
annual temperature variation beneath it.

(c) In the Bornholm deep in the Baltic, no noticeable tidal current and a strong
increase in salinity with depth (strong density stratification during the whole year)
so that the lower layer is isolated and shows no annual temperature variation. See
p. 115 and Munk and Anderson (1948 on the theory of the thermocline).

The annual heat budget of a limited water mass can also be calculated without diffi-
culty if sufficient observational data are available. A number of calculations of this type
have been made for lakes and similar calculations have been carried out for more or less
enclosed seas. According to O. Pettersson ( 1 896) the Baltic gives off" 1 37-500 kg cal/m-
from August to November and a further 385-500 up to March, in total about 523-000

The Three-dimensional Temperature Distribution and its Variation in Time 117

kg cal/m^. The annual heat budget for the Ionian Sea has been calculated by Hann
(1906, 1908) as about 371,000 kg cal/m^; for the sea south of Cyprus he found 426,000,
for the Bay of Naples 432,000 and for the Black Sea 482,000 kg cal/m^. In the polar
regions the annual heat budget is much smaller. Malmgren (1927) has made corre-
sponding calculations for the North Polar Basin; he estimated that the atmosphere
received 68,000 kg cal/m^ from the sea annually. This mean annual value was obtained
from the difference between the loss of 76,700 kg cal/m^ from September to April
and a gain of 8,700 kgcal/m^ from June to August. See Deitrich (1950) for a dis-
cussion of the annual variation of heat content in the English Channel.

4. The Vertical Distribution of Temperature in the Ocean

Figure 52 shows the vertical temperature distribution for a series of oceanographic
stations along a meridional cross-section through the middle and central parts of the
Atlantic. The general and common characteristics of the vertical temperature distribu-

Temperature, °C

0 4 8 12

Fig. 52. Vertical temperature distribution at a series of stations along a meridian in the

Atlantic Ocean :

'•Will. Scoreby'

554

63° 20' S.

17°

23' W.

1 5143 m

5. ii. 1931

"Meteor" 58

48° 30' S.

30°

O'W.

4989 m

7/8. X. 1925

"Meteor" 83

32° 9'S.

25°

4'W.

4506 m

29. xi. 1925

"Meteor" 170

22° 39' S.

27°

55'W.

5454 m

9. vii. 1926

"Meteor" 191

9° 7'S.

2°

2'W.

1 4533 m

9/10. ix. 26

"Meteor" 212

0° 36' N.

29°

12' W.

3773 m

19. X. 1926

"Meteor" 283

17°53'N.

39°

19' W.

5748 m

22/23. iii. 1927

"Dana" 1376

33°42'N.

36°

16' W.

10. vi. 1922

"Armauer Hansen" 17

58° O'N.

11°

O'W.

1860 m

29. vii. 1913

10.

"Fram" 29

78° I'N.

10' E.

, 1075 m

22. vii. 1910

tion thereby stand out clearly. Conditions in the other oceans are also essentially the
same. The typical curve for the vertical temperature distribution in the open ocean is ana-
thermic, that is it shows a decrease of temperature with increasing depth, though this
decrease is not uniform. In latitudes between about 45° S. and 45° N. the thermal

118 The Three-dimensional Temperature Distribution and its Variation in Time

stratification of the sea is characterized by two principal layers. The upper layer ex-
tends from the surface down to about 600-1000 m and is termed the oceanic tropo-
sphere; its uppermost part down to about 100 m is subject to the direct influence
of the atmosphere. This is the layer of diurnal and annual convections originating
at the surface and it shows the strongest mixing due to the effects of the wind and
waves ; it can be designated as the layer of surface disturbances. The troposphere
shows the strongest temperature decrease with depth and in low and middle latitudes
forms an upper warm layer of water overlying the cold water masses underneath and
separated from them by a more or less sharply marked thermocline.

Table 47. Mean vertical temperature (°C) distribution in the three oceans
between 40° N. and A0° S,.

Atlantic Ocean

Indian Ocean

Pacific Ocean

Mean

Depth

(m)

100 m

^°

A^°l
100 m

^°

^^7
100 m

^°

A^°l
100 m

200

22-2

21-8

21-3

100

17-8

2-2

18-9

3-3

18-7

3-1

18-5

2-8

200

13-4

4-4t

14-3

4-7t

14-3

4-4t

140

4-5t

400

9-9

1-8

110

1-6

9 0

2-6

100

600

1-5

8-7

1-2

6-4

1-2

7-4

1-3

800

5-6

0-7

6-9

0-9

5-1

0-65

5-9

0-75

1000

4-9

0-35

5-5

0-7

4-3

0-4

4-9

0-5

1200

4-5

0-20

4-7

0-4

3-5

0-4

4-2

0-35

1600

3-9

015

3-4

0-3

2-6

0-2

3-3

0-22

2000

3-4

012

2-8

015

2-15

2-8

012

3000

2-6

008

1-9

009

1-7

005

007

4000

■ •8

008

1-6

003

1-45

003

1-6

005

t Maximum

The lower part of the thermal stratification is the oceanic stratosphere which extends
from the bottom of the troposphere (thermocline) down to the sea bottom; to it belong
the major water masses of the deep sea which are characterized by the very small changes
in temperature both in horizontal and vertical direction. Table 47 presents the mean
vertical temperature distribution in the three oceans for latitudes between 40° N.
and 40° S. and also the vertical temperature gradient at each depth in degrees
per 100 m. The approximate limits between the zone of disturbance, troposphere and
stratosphere are indicated in Fig. 53. This twofold subdivision in the thermal structure
of the ocean is limited to the tropical and subtropical parts of the ocean. As is shown in
Fig. 52 the troposphere becomes less well developed towards higher latitudes and the
stratosphere comes closer to the sea's surface. In the subarctic and subantarctic regions
(polewards of the oceanic polar front, see Chap. XIX) the troposphere disappears
and the cold-water masses of the stratosphere extend generally to the surface. The
water masses of the troposphere lie on top of and are embedded in the cold-water
mass of the stratosphere in tropical and subtropical areas, but thin out and disappear
in higher northern and southern latitudes.

Because of the decrease of salinity with depth it can be expected, just for reasons of
stability, that the temperature must also decrease with depth. Solar radiation is con-
verted into heat in the upper layers and from here the heat spreads rapidly downwards.

The Three-dimensional Temperature Distribution and its Variation in Time 119

0 2

0 2 4 6 8 10 12 14 16 18 20 22

0 2 4 6 8 10 12 14 16 18 20 22
Fig. 53. Mean vertical temperature distribution in the three oceans.

The temperature distribution of the ocean must be regarded as quasi-stationary and
this leads to the deduction that the vertical temperature distribution is a phenomenon
closely connected with the oceanic circulation. Assuming that there was no motion in
the very deep ocean this vertical temperature distribution could not be understood.
Humboldt (1816) emphasized at an early date that the low temperature at great
depths in the tropical ocean can only be explained by assuming an equatorward flux
of cold-water masses originating in high latitudes.

{a) The Oceanic Troposphere

In general, the troposphere shows a well-developed subdivision into three parts.
In the top layer the vertical differences in temperature and salinity are very small —
so frequently that this top layer can be regarded as homogeneous. Its thickness is
seldom greater than 100 m. In the Atlantic an isothermal surface layer (tempera-
ture gradient <0-015°/m) is present only in the region between about 35° S. and 25° N.
polewards from these limits the isothermal stratification is slowly destroyed and the
effect of the seasons begins to predominate (disturbance zone). Table 48 presents mean

Table 48. The quasi-isothermal top layer in the Atlantic Ocean

Total no. of stations

Mean geographical
position

24 °S.
16° W.

15° S.

15° W.

9°S.
17° W.

0°S.

22° W.

8°N.
23° W.

18° N.
36° W.

Depth (m)
0
25
50

20-36
20-32
20-38
20-37
20-30

24-10

24-44
24-45
23-46

Temp
24-40
24-36
24-28
23-79

.(°C)
26-50
26-43
26-28

25-80
25-82
25-43
24-55

22-78
22-86
22-91

22-77
17-02
13-42

22-65

100

20-65
17-10

20-32
14-60

19-77
12-98

22-50

150

17-72

20-22

1 20 The Three-dimensional Temperature Distribution and its Variation in Time

conditions at several stations in the central part of the Atlantic. In the subtropics
(30°-20°S. and 20°-25°N.) the isothermal layer extends down to about 100 m,
but is more shallow in the tropics and in regions close to the equator (in the west
about 75 m and in mid-latitudes 50 m or less). Off the African coast, especially in the
Gulf of Guinea, the thickness decreases to 25 m or less, and in the regions with cold
water upwelling it is entirely absent. Underneath the top layer there is a strong tem-
perature decrease that continues, gradually weakening, down to the lower limit of the
troposphere. The maximum of vertical temperature gradient (thermocline) is generally
found between 100 and 200 m, with a mean value of nearly 5°C per 100 m. The
meridional variation of the depth of thermocline is shown in Table 49 (Fig. 54).

Table 49. Meridional variation of the depth of thermocline in the Atlantic Ocean
(Mean values for the entire ocean)

Latitude
Depth (m)

20° S.
141t

15°
121

10°
108

5°S.

0°
69*

2-5 °N.
83t

5°
81

10°

53*

15°
89

20°
160

25° N.
195*

* Minimum; f Maximum

\ /]

100

150

200

20° S 10° 0° 10° 20° N

Fig. 54. Meridional distribution of the depth of thermocline in the Atlantic.

The thermocline rises steadily from a depth of 150 m in the subtropics to minimum
values in the equatorial regions. Approaching the equator from the Southern Hemi-
sphere a minimum of about 70 m is reached directly at the equator; however, coming
from the north the minimum (about 55 m) already shows in 10° N. Between these two
highest locations the thermocline drops about 1 5-20 m to a deeper level (approx. 80 m)
at 2-5° N. These changes in level are rather characteristic for the entire width of the
ocean and due to dynamical reasons are associated with the zonal oceanic circulation of
the equatorial water masses (see Chap. XVIII and XIX). The intensity of the thermo-
cline is greatest in the equatorial areas, where it has a mean value greater than
0-4 °C/m. An actual transition layer (temperature gradient >0-rC/m) properly
speaking only occurs between 15° N. and 15° S.; on either side of this belt the
gradient falls rapidly toO-05° C/m or lower and the transition layer shows only as
an intensification of the vertical temperature gradient.

The Three-dimensional Temperature Distribution and its Variation in Time 121

Table 50. Heat transport downwards assuming a temperature gradient of 1 °C/ 1 00 m

Vertical exchange coefficient (/Ij g cm~^ sec"^) 20 10 5 2-5 1

Heat amount (g cal cm-2 day-i) 172 86 43 21-5 8-6

Beneath the thermocline from about 200-300 m the water masses of the sub-
troposphere are remarkably constant in their nature and geographical distribution.
The vertical temperature gradient in these waters rapidly decreases with depth and
gradually changes its magnitude into that of the stratosphere. Considerable amounts
of heat are transported by dynamic convection through the layer immediately be-
neath the almost isothermal top layer to the layer below. Table 50 gives an idea of the
quantities of heat involved; it assumes a mean temperature gradient of 1°C/100 m.
These amounts of heat are surprisingly high. Even for small values of A^, the down-
ward heat flux amounts to 10-40 gcal cm "May ""^. Since there is always a tem-
perature gradient, this raises the very natural question of where all this heat goes to.
In the lower layers of the troposphere the temperature gradient is again smaller and
therefore the downward heat flux becomes smaller again in the middle layers of the
troposphere ; the accumulation of heat in these layers should soon destroy the vertical
temperature gradient and thus also the thermocline. It must therefore be true that the
vertical temperature gradient in the troposphere can only be maintained if the lateral
influx of colder water compensates the flow of heat from above and indeed the heat
from above and the horizontal advection must compensate each other exactly. The
vertical temperature distribution in the troposphere is thus maintained in a stationary
state by the oceanic circulation (Defant, 1930).

The cause for formation of the thermocline below an almost isothermal top layer
in the tropics and the subtropics is therefore as follows: The top layer is certainly
more or less in thermal equilibrium with the atmosphere above. The lower tempera-
tures of the lower subtroposphere and of the stratosphere are essentially of polar
origin; as they flow towards the equator these water masses mix with warmer water and
thereby gain heat, but are continually renewed and are thus kept at a relatively low
temperature. It would be expected that the diff"erence between the high temperature
at the top and the low temperature of the deeper layers would give rise to a roughly
linear vertical temperature gradient in the middle layer; instead a homogeneous top
layer is formed and the transition to the lower temperatures of the subtroposphere
takes place abruptly in a well-developed transition layer (thermocline).

The explanation of this thermal stratification in the tropics and the subtropics lies
in the same circumstances that give rise to the summer transition layer in lakes as well
as in the ocean. The turbulence induced by the wind and the waves will slowly trans-
port the heat from the upper layers downwards and the temperature diff"erences thus
formed will work their way down into deeper and deeper layers. However, further
rise in temperature in the top layers will also increase the vertical density gradient.
The downward transfer of heat from above by turbulence will cease when the increase
in vertical stability diminishes the intensity of the turbulence. If the vertical density
gradient is very strong the turbulence of the flow cannot overcome the great stability
of the stratification and a transfer of heat to a deeper level through the thermocline
can no longer occur. In the top layer the turbulence leads finally to a complete

1 22 The Three-dimensional Temperature Distribution and its Variation in Time

equalization of temperature thus forming an upper isothermal top layer. Beneath this,
at a definite constant level, lies the thermocline, which acts as a barrier for all turbulent
processes. The important point in the explanation of the formation of tropical and
subtropical thermoclines is the exclusion of turbulence and their consequences in a
fixed depth due to the increase of the vertical density gradient above a critical value.
The condition for the reduction and final elimination of the turbulence in a non-
laminar flow is that the dimensionless quantity (Richardson number):

{glp){hpidz) Ab
{dujdzY -^ At'

In this relation u is the basic velocity of the turbulent flow, /Ib is the exchange coefficient
for the flow momentum (apparent viscosity) and At is the exchange coefficient for
density differences (temperature and salinity). In the ocean the ratio between these
two quantities is between about 5 and 20 (see p. 103). In the thermocline of the equa-
torial region of the Atlantic the quantity Spjdz is of the order of 5 X 10"* for a vertical
interval of 20 m. In drift currents dujdz can be taken as about 10 cm/sec for every
20 m. The left-hand side of the above inequality is thus 100, which is considerably
more than the value of the right-hand side. With such a stratification the turbulence
in a current cannot be maintained (Defant, 1936),

The basis of the theory for the formation of the thermocline has been given by
MuNK and Anderson (1948). They have shown that the sharp transition between the
top layer with mixing and the thermocline can be explained theoretically on the as-
sumption that the eddy coefficients are a function of the vertical stability and of the
wind shear. This theory gives a value for the depth of the thermocline that is some-
what too sm.all but it is of the correct order of magnitude. This depth depends on the
wind velocity, on the latitude, on the heat flux and on the [r^Sl-relation in that order.
This theory undoubtedly penetrates deeply into the important processes that control
this phenomenon but it does not yet completely satisfy all points. Experimental
investigation and systematically planned observations would very probably improve
the basis of the theory.

(b) The Oceanic Stratosphere

The vertical temperature differences in the very deep layer of the oceanic strato-
sphere are small. Here also the distribution is almost everywhere anothermic; however,
the temperature gradient at depths below 1000 m falls rapidly to values less than
0-4°C per 100 m, at 2000 m it is at the most O-TC and at 3000 m and below it is
barely 0-05 °C/ 100 m. Departures from this anothermic distribution are found only
in the Western Atlantic (Brazilian and Argentinian basins) and in the south-western
Indian Ocean where at a depth of 1300-1600 m there is a very weakly marked tem-
perature inversion, a phenomenon of particular importance for the oceanic circulation
of these oceanic spaces. Table 51 shows particularly well-developed inversions at some
"Meteor" stations. Inversions such as these occur only rarely in the eastern half of the
South Atlantic and are very weak. They appear to be due to long-term changes asso-
ciated with aperiodic variations in intensity of the deep-sea circulation (Merz, 1922;
WiJST, 1936, 1948).

The Three-dimensional Temperature Distribution and its Variation in Time 123
Table 51. Temperature inversions in the western Atlantic (°C)

"Meteor" station

Depth (m)

800

900

1000

1200

1400

1600

1800

2000

170: 22-6° S., 27-9° W.
158: 15-9° S., 300° W.
201: 9-5° S., 300° W.

4-55
403
4-125

3-91
3-70
3-91

3-53

3-605

3-79

305

3-79t
3-97

302

3-78
4-13t

3-465t

3-55

3-86

3-45
3-34
3-54

3-30
3-14
3-31

t Maximum

Also in a horizontal direction the temperature differences in the stratosphere are
small. The temperature distribution here must certainly be due to the stratospheric
circulation which starts from the locations where the stratosphere extends up to the
surface, that is in the polar and subpolar regions where it is in direct contact with the
atmosphere. The water masses that sink in these places, where the major convection
processes (see p. 97) originate, spread out very largely in a quasi-horizontal direction
towards the equator to fill up the greater part of the space underneath the troposphere
of the tropics and subtropics, and are thereby subjected to considerable lateral mixing
at the same time.

SvERDRUP (1938) has pointed out that the stratospheric temperature distribution
can be mainly explained on the assumption that there is extensive lateral and vertical
mixing of the water masses. This mixing takes place along the isopycnic surfaces that
rise towards the surface in the polar and subpolar parts of the oceans. Figure 55 shows
that the temperature distribution in a meridional cross-section through the Atlantic
below 1000 m can be interpreted roughly as due to the effects of this lateral and
vertical mixing; the theoretical isotherms calculated from the equation on p. 108
taking Ax : Ay as 6 x 10^ follow a similar course than the observed isotherms. The
temperature distribution in the Atlantic asymmetric to the equator is partly due to
the effects of an inflow of warm water from the Mediterranean and partly due to the
strong cooling effect of the Antarctic. It cannot be doubted that mixing along the
isopycnic surfaces in the oceanic stratosphere is of very considerable importance
in the distribution of the oceanographic elements.

Since sea-water is compressible, although only slightly, the pressure changes
undergone by a small mass of water in the ocean must be accompanied by adia-
batic changes in temperature which can be significant for oceanographic problems.
Nansen (1900, 1902) first drew attention to the thermal effects of the compressibility
of sea- water. If a mass of water is raised from a given depth to a shallower one, will be
subjected to less pressure and will expand, performing work against the external
pressure, and the water will be cooled by a definite amount. Analogous conditions will
apply for a water mass which sinks ; its temperature will increase. Since the compressi-
bility of water is not large these temperature changes will remain only in hmits ; how-
ever, since the vertical temperature gradient in the deeper layers is extremely small,
these adiabatic ejfects must be taken into account.

The adiabatic temperature change Si^ for a displacement from a depth /z^ to a depth

1 24 The Three-dimensional Temperature Distribution and its Variation in Time

00091-

OOOt^l

000 a

0009

-) -9,

a O
u • —

V =5

"3 .i2

i2 o

^.2

y t

(\J

— o

ILI

■-

O r-

2 y

^ S

i^eq

C.2

y 3

O X)

—

O-C

O to

~1 3

< «

•5 cu

o
o

O
O

ro
• 'mdSQ

o
o
o

o
o

?r 3
c ^

2^
o

^ o

3 ""

.-3 60

-J o

IT) w

The Three-dimensional Temperature Distribution and its Variation in Time 125

h^ can be calculated using a formula derived from the energy principle by Lord Kelvin j
v^'hich in c.g.s. -units take the form:

''" Ta*g

r dz,

hi ^v'

where T is the absolute temperature of the water, a* is its coefficient of thermal ex-
pansion, Cp is the specific heat at constant pressure, g is the gravitational acceleration
and J is the mechanical equivalent of heat (4-1863 x 10' erg/cal).

The adiabatic temperature change hd for a displacement from a depth h^ to a depth
//o is thus dependent on the coefficient of thermal expansion and on the specific heat
of sea-water, which are both effectively dependent on the temperature, the salinity
and the pressure (see p. 49).

After solving the above equation, Ekman (1914) has presented numerical values
which allow an easy determination of the adiabatic effects for sea-water. Helland-
Hansen (1930) later prepared from these values tables giving directly the adiabatic
heating and cooling in sea-water of o^ = 28-0 (corresponding to a salinity of 34'85%o)
when raised from a given depth to the surface with a given temperature; a further
table gives the adiabatic temperature change for the upper 100 m for salinities be-
tween 30-0%o and 38-0%o. With these tables or the corresponding diagrams, any adia-
batic change can be determined without difficulty. Table 52 is extracted from these
tables.

Example: at a depth of 9788 m (Philippine Trench) a temperature of 2-60° C was
measured and a density o- = 28. What would be the temperature of the water for an
adiabatic ascent to the surface? Table 52 gives, by interpolation, a T-change at 2-60°
of — M37°C for 9000 m; for 10,000 m the change would be —1-319° and this for
9788 m —1 -280-0. If the water at 9788 m rises to the surface there will be an adia-
batic temperature change from 2-60°C to 1-32°C.

The temperature of a water mass after being moved adiabatically to the surface
is known as the potential temperature. It is given hy d = d -\- 8§. If the vertical
stratification of the sea were such that the salinity were constant, so that the density
would only depend on the temperature, then the equilibrium state ofthe sea could be
shown by the vertical distribution of the potential temperature in the same way as in the
atmosphere. Complete mixing of the water masses in vertical direction would eliminate

t This above equation can be derived without difficulty from the first and second laws of thermo-
dynamics. If the state of a body is defined as a function of the temperature T and the pressure p,
then

T da
dQ = c,dT-j^dp.

Taking the definition of the coefficient of thermal expansion (see p. 48) as

1 da
-^ = «*

a ct

and the static equation as dp = gp dz then for an adiabatic process (dQ = 0) with pa = \ and
/& = JT

8 »= f dz

or for the interval from h^ — h^ the above formula is derived.

1 26 The Three-dimensional Temperature Distribution and its Variation in Time

Table 52. (A) Adiabatic cooling {in 0-01 °C) resulting from an

ascent of a water particle of temperature d'm up to the sea surface

(a = 28-0 ; S = 34-85%o)

i>™(°C)

Depth

(m)

-2

1000

2-6

4-4

6-2

7-8

9-5

12-4

2000

7-2

10-7

14-1

17-2

20-4

23-3

26-2

3000

13-6

18-7

23-6

28-2

32-7

37-1

41-2

4000

21-7

28-4

34-7

40-6

46-3

51-9

57-2

6000

42-8

52-2

61-1

69-4

—

8000

—

81-5

92-5

102-7

—

10,000

—

115-7

128-3

140-2

—

(B) Adiabatic temperature change (in 0-01 **€) for the upper
1000 m at different salinities

»(°C)

S%o

3-5

7-0

10-3

13-2

16-1

18-9

3-9

7-3

10-6

13-5

16-4

19-1

4-3

7-7

10-9

13-8

16-6

19-3

4-7

8-1

11-2

14-1

16-9

19-6

5-1

8-4

11-6

14-4

17-2

19-8

all temperature differences except those due to adiabatic effects. The temperature of
each depth would be fixed by purely adiabatic displacements of water from the surface
or from the bottom to the given depth and in that way the vertical distribution of
temperature would remain invariable. In this case a mass of water from the surface
would be subjected neither to a force upwards nor to a force downwards, but would
always be in equilibrium with its surroundings (indifferent equilibrium). In a vertical
direction the potential temperature within it would be constant. The vertical distribu-
tion of temperature in such a case for some initial values at the sea surface is shown in
Table 53. In neutral equilibrium there is thus a slight increase of temperature with
depth which does not reach a temperature of 1-5 °C in the 10 km depth.

Table 53. Vertical temperature distribution of indifferent equilibrium

Potential
temperature

Depths (km)

CC)

0-00

5-00

10-00

0-045

5-087

10125

0-109

5-191

10-265

0-192

5-312

10-419

0-293

5-448

10-587

0-412

0-698

1-044

If the vertical temperature gradient is greater than the adiabatic, i.e. if the potential
temperature calculated from the temperature in situ increases with increasing depth,

The Three-dimensional Temperature Distribution and its Variation in Time 1 27

then the equilibrium state is unstable in the vertical. If a small water mass in such a
thermal stratification is displaced downwards, it will remain colder than its surround-
ings in spite of adiabatic heating, and it will be forced down further and further from
its initial position. If it is displaced upwards then it will remain warmer than the
surroundings and will therefore continue to rise. If, on the other hand, the vertical
temperature gradient is less than the adiabatic, particularly if the temperature de-
creases with depth, then the potential temperature will also decrease with depth and
the stratification is stable.

Table 54. Vertical distribution of potential temperature (°C) below 3000 m for several
stations in the western and eastern troughs of the Atlantic Ocean

Western trough

]

Eastern trough

North

Depth

Argentina

Brazil Basin

America

Antarctic

Cape Basin

Cape Verde

(m)

Basin

Met. 56

Met. 249

Dana 1356

Met. 129

Met. 77

Met. 264

48-4° S.,

50° S.,

300° N.,

58-9° S.,

34-0° S.,

10-2° N.,

42-6° W.

26-4° W.

59-6° W.

4-9° E.

30° E.

26-6° W.

3000

+ 1-40

+2-44

+2-66

-0-60

+207

+2-46

3500

+0-96

+2-28

+2-32

-0-73

+ 1-73

+2-22

4000

+0-33

+ 1-66

+2-03

-0-82

+0-94

+2-06

4500

-005

+0-48

+ 1-85

-0-85

+0-68

+ 1-92

5000

-0-27

(+0-25)

+ 1-66

-0-86

+0-60

+ 1-84

5500

-0-27

—

+ 1-61

-0-88

—

+ 1-77

The vertical temperature distribution present in the ocean is such that the stratifica-
tion, in so far as it depends on the temperature, is stable. In the oceanic troposphere
the temperature decrease is so large that, in spite of the vertical decrease in salinity,
the equilibrium state remains quite stable. In the upper layers of the stratosphere
the stratification is still stable, however, it becomes continuously less stable with in-
creasing depth. Table 54 shows the vertical distribution of the potential temperature
below 3000 m for several stations in the eastern and western troughs of the Atlantic
Ocean which show these conditions rather clearly. The same is usually also found in
the open sea of the Indian and Pacific Oceans.

At very great depths, below about 4500 m, especially in the more or less extended
deep-sea basins, the vertical temperature distribution approaches the adiabatic and
may even exceed it a little, so that there is an indiff'erent stratification at great
depths or sometimes it may even be slightly unstable. It is principally in the deep-
sea trenches of the Pacific and Indian Oceans that this occurs. In these there is nearly
always temperature increase, but it seldom exceeds the adiabatic gradient and if it
does then only by very little. Such a condition of indiff'erent stability is formed only
when there is an almost complete separation of the water mass from the surrounding
waters. More or less fully enclosed deep inland seas such as the individual basins
of the European Mediterranean and the North Polar Basin show this phenomenon to
a marked degree in their deeper parts. The classic example of conditions in a deep-sea
trench is the vertical temperature distribution in the Philippine Trench (Schott,
1914; SCHULZ, 1917; Wust, 1937; van Riel, 1934; Schubert, 1931). According to

128 The Three-dimensional Temperature Distribution and its Variation in Time

more recent calculations from the observations made in this trench there is no in-
stability in the deepest layers as was previously supposed. Table 55 presents the data
obtained in this case.

Table 55. Vertical temperature distribution in the Philippine Trench: ""Will. Snellius"
Exp., Stat. 262 (9° 40-5' N., 126° 50-5' E.)

Depth (m)

1455

1970

2470 2970

3470

3970 4450

5450

6450 , 7450

8450

10035

Temperature

(^C)

3-205 2-27 1-825

1-66

1-585*

1-595

1-64

1-78

1-925

2-075

2-23

2-475

Potential

temp. (°C)

3 095

2-13

1-65

1-44

1-31

1-26

1-25

l-26t

1-25

1-24

1-22

116

Change over

500 m

(orc)

_9.4 _4.8 -2-1 -1-3 -0-5 -01 +0-05 -005 -005 -005 -0-1

Salinity (%„)

34-58

0-605

0-64

0-66 0-67 0-67

0-67

0-67 0-67

0-685

0-695

0-67

* Minimum; j Maximum

Between 3500 and 10,035 m with almost constant salinity there is an increase in
temperature from 1-58 to 2-47°C, an increase of 0-89°C; this increase is, however,
less than the adiabatic one ; the stratification is thus still stable but very close to the in-
different equilibrium. At a level of 5500 m there exists a small anomaly because a
thin layer of water, with a warmer potential temperature (1-26°C), is situated under-
neath another layer with a colder potential temperature (1-25°C). The difference is,
however, only small. The stratification here is thus very close to a vertically unstable
state. However, if the salinity would decrease only a little more with depth the weakly
stable temperature stratification could be changed by the salinity into an indifferent
or even into a slightly unstable one.

It was at first supposed that the almost adiabatic or slightly superadiabatic tempera-
ture gradient, in the deep-sea trenches and the deep troughs of the major oceans, was
due to a heat gain from the solid Earth. The heat transferred from the interior of the
Earth to the lowermost water layer per second is

Q = -2-1 X 10-«gcal/cm2 (see p. 88).

This heat amount would accumulate in the layer very close to the sea bottom, until
such a temperature gradient is formed that the incoming heat per unit time would
equalize the heat transfer to the layers above. If the water mass were to be completely
motionless, then according to the calculations of Schmidt (1925), the stationary
temperature gradient would be determined by the heat entering the layer from the
Earth and by the coefficient of thermal conductivity, so that in this case there would be
a temperature decrease away from the bottom of 1-5°C in 10 m.

dO 2-1 X 10-« , ^ ,^3 _,

-, = y-. :r,r-„ == 1-5 X 10-=^ °C/cm.

dz 1-4 X 10-^

Thus, in a deep-sea trench below 5000 m the temperature should rise linearly due to
the heat transferred from the Earth to the water, and at the bottom (10,000 m) would
be over 700°C. Since this does not occur it must be concluded that even the deepest

The Three-dimensional Temperature Distribution and its Variation in Time 129

layers are not at complete rest, and that due to the exchange produced by turbulent
motions in these layers the heat is more rapidly dissipated than by normal conductivity.
An exchange of about 4 g cm-^ sec~^ would be sufficient to account for the
observed slightly superadiabatic temperature gradient. This appears, however,
not entirely conclusive. Even when the water masses in these more or less enclosed
deep-sea troughs and trenches do not participate in the general horizontal circula-
tion of the deep sea and can therefore be regarded as motionless in horizontal direc-
tion, there may still occur vertical convection currents produced by the continuous
influx of heat from the Earth which will carry this heat to the layers above. Such a
convection will be effective if there is the smallest vertical instability. Once such
instability exists in the bottom layer there will be a steady interchange of small water
quanta rising and sinking, and this convectional circulation will be maintained by the
steady inflow of heat through the sea bottom. In the water masses above an adia-
batic temperature gradient will be established; a gradient greater than the adiabatic
can, however, form only in the very bottom layer, though even here it will be scarcely
possible to detect it by physical measurement. It is required here in order to maintain
the vertical circulation against the internal viscosity. This might be the cause that the
water masses of the deep-sea trenches and the deep basins in the ocean show a vertical
stratification approximating closely indifferent equilibrium state.

{d) The Vertical Temperature Distribution in Adjacent Seas

While a steady decrease in temperature with increasing depth is characteristic for
the open oceans, in adjacent seas connected with the open ocean over shallow sills
the temperature below a certain depth is almost constant no matter how deep they
may be. The adjacent seas can be divided into two groups according to their tempera-
ture stratification : the first includes all those adjacent seas where the surface water in
winter cools to a temperature which is lower than that of the open ocean at the greatest
depth at which they are in communication (sill depths). Provided there is an almost
homo-haline structure in these adjacent seas, the autumn and winter convection causes
the cooled surface water to sink to the bottom, and the deeps in these adjacent seas
are thus filled with water masses at approximately the lowest surface temperature
occurring during the coldest month of the year. The deep layers in this show roughly
the winter temperature of the region concerned, provided the convection is not pre-
vented from reaching the greatest depths by irregularities in the thermo-haline struc-
ture of the surface layers, for instance, by a layer of low salinity.

Examples of this type of adjacent sea are the Red Sea and the European Mediter-
ranean. In the first case, in the Straits of Bab-el-Mandeb (north of Perim island), the
sill depth is 1 50 m; in the second, in the Straits of Gibraltar, about 350 m. In the
Mediterranean during the summer there is a pronounced anothermal stratification
in the upper layers, while depths below about 300-400 m are essentially homo-
thermal. Towards the end of the winter this homo-thermal state extends upwards to
the surface. The temperature of this deep layer is thus about 12-9-1 3-2 °C in the
Balearic Basin and in the Tyrrhenian Basin, and about 1 3-6-1 3-9°C in the Ionian
Basin and in the eastern basin near the Syrian coast. The northern Adriatic Sea shows
values near to 12°C. These temperatures are all in good agreement with the winter
temperatures in these regions (Table 56).

130 The Three-dimensional Temperature Distribution and its Variation in Time

Table 56. Vertical distribution of temperature and salinity in the
European Mediterranean

Tyrrhenian Sea

Ionian Sea

"Dana" 4119 (30.V. 1930)

"Thor" 144 (23.vii.1910)

Depth (m)

40° 13' N., 12° 6' E., 3400 m

34°31'N., 18°40'E.; 3340 m

rrc)

5 (%„)

r(°C)

S (%o)

20-0 37-72

26-05 38-49

17-36 37-80

22-50 38-13

14-79 1 38-00

17-28 38-26

100

13-81

38-30

15-40

38-35

150

14-09

38-50

14-66

38-64

200

14-12

38-60

14-41

38-78

400

14-13

38-69

13-96

38-77

600

13-76

38-61

13-76

38-72

1000

13-19

38-49

13-58

38-66

1500

13-06

38-46

13-55

38-64

2000

13-04

38-44

13-56 1 38-64

3000

13-21 38-41

Bottom

(3200 m)

13-30

38-42

(3000 m) 13-69

38-64

During the autumn and winter tlie deep water forms at the surface and is carried
by the convection to the deep basins. This is not influenced through the Straits of
Gibraltar, since the bottom current through the straits carries water out from the
Mediterranean and the influence of the upper current on salinity and temperature does
not reach very far to the east.

Conditions in the Red Sea are similar (see Table 57, Riel, 1932).

Table 57. Vertical distribution of temperature and salinity in the Red
Sea and in the Gulf of Aden. ''Will. Snellius" Exp., April 1929

Depth
(m)

Red Sea

St. 18

15° 52' N., 44° 43' E.

Straits of

Bab-el-Mandeb

St. 19

13° 27' N., 42° 51' E.

Gulf of Aden

St. 20

12° 55' N., 45° 48' E.

r(°c)

S(%o)

T{°0

S (%o)

r(°c)

S{%o)

100

150

250

500

600

700

900

1000

26-70
26-10
25-99
22-51
21-94
21-66
21-59
21-58
21-60
21-63
21-66

37-07
37-11
37-42
40-27
40-46
40-57
40-61
40-57
40-60
40-60
40-60

Bottom 1030 m

27-60
27-41
27-28
24-53

36-12
36-33
36-36
38-57

28-80

125 m
22-80 39-98

Bottom 135 m

36-25

25-36

36-05

22-70

35-87

18-12

35-52

14-76

35-45

14-89

35-36

13-50

35-24

12-51

35-22

10-26

35-88

The Three-dimensional Temperature Distribution and its Variation in Time 131

Below about 300 m down to the greatest depths it is filled with a water mass at a
temperature between 21-5°C and 21-6°C. The deep water has its origin at the surface
in the northern half of this sea, where in March and April the v/ater temperature is
21-5°C combined with salinity values of 40-5-40-7%o increased by evaporation. The
currents present definitely exclude any influence from conditions outside the open
straits in the south.

With this group can be included the temperature distribution in the deeper layers
in the Norwegian Sea (from 1000 to 3500 m approximately homo-thermal, —0-8 to
— 1-3°C and 34-9%o). Presumably this water mass must be formed at the surface to
the north of Jan Mayen.

The second group of adjacent seas belongs exclusively to the warmer zones, where
the surface temperature during the whole year is so high that the temperature at the
sill depth is the determining factor for the thermal structure of the sea below the sill
depths. Only oceanic water has access in this case to the deeper layers below sill
depth. The sinking of oceanic water into the enclosed space produces a potential
temperature extending to the bottom, that is determined by the potential temperature
of the open ocean at the level of the sill. This phenomenon is in many cases so marked
that inversely the sill depth can often be deduced from the vertical temperature distri-
bution in the adjacent sea,

A characteristic example of this second group is the quasi-homo-thermal structure
of the water masses in the Australian-Asiatic deep-sea basins beneath the depths of
the sills over which they are connected with the Pacific Ocean or with the neighbouring
basins. An accurate and detailed investigation of these conditions based on the ob-
servations made by the "Willebrod SneUius" Expedition has been made by Riel
(1934), Table 58.

Table 58. Vertical distribution of temperature and salinity in the
Australian-Asiatic Basins {""Will. Snellius" Exp.)

Sulu Sea

Celebes Sea

Banda Sea

7°N., 120° E.,

3°N., 121° E.,

7°S., 128° E.,

Depth

Sept. 1929

Apr. 1930

(m)

TCO

S(%o)

r(°C)

SCYoo)

TCO

S(%o)

27-8

33-46

28-4

34-22

28-4

33-48

27-75

33-59

27-33

34-33

27-07

34-20

100

24-26

34-32

24-41

34-68

21-42

34.52

150

18-66

34-40

20-44

34-81

17-46

34-60

200

15-25

34-48

17-26

34-70

13-71

34-56

400

11-50

34-50

8-99

34-42

8-83

34-57

600

10-53

34-47

6-90

34-52

6-62

34-55

800

1015

34-45

5-54

34-52

5-71

34-59

1000

10-08

34-46

4-49

34-55

4-70

34-59

1500

10-09

34-47

3-78

34-58

3-71

34-59

2000

10-14

34-47

3-61

34-57

3-24

34-61

3000

10-28

34-46

3-60

34-58

3-06

34-61

4000

—

3-72

34-59

—

Bottom

10-42

34-45

3-77

34-59

310

34-61

Bottom 3950 m

Bottom 4773 m

Bottom 3308 m

132 The Three-dimensional Temperature Distribution and its Variation in Time

The Sulu Sea between Borneo and the Philippines is connected in the north with the
Pacific through a sill with a maximum depth of about 400 m. Below this depth the
vertical temperature gradient becomes very small and down to the greatest depth at
approximately 5580 m the temperature remains almost constant (minimum 10-07°C
at 1225 m, rising to 10-42°C at the bottom). The deep basin of the Celebes (greatest
depth 6220 m) has an almost constant temperature below 1400 m (sill depth at 1400 m
in the Kawio Strait). The broad Banda Basin has a sill depth of 3130 m and in the
northern part shows a temperature minimum of 3-04 °C at 2990 m, in the southern part
3-06 °C at 2720 m.

Similar conditions are also present in the American Mediterranean. The main
morphological structure consists of three major basins: the Gulf of Mexico, the
Yucatan Basin with the Cayman Trench and the Caribbean Basin (Parr, 1932,
1937, 1938; see also, Dietrich, 1937, 1939). Table 59 shows the vertical distribution
of temperature and salinity at three stations in the three major basins of this adjacent
sea. Figure 56 shows several characteristic vertical temperature distributions for four
adjacent seas.

Table 59. Vertical distribution of temperature and salinity in the
American Mediterranean

Gulf of Mexico

Cayman Trench

Caribbean Sea

"Mabel Taylor" 1104

"Atlantis" 1570

"Atlant

is" 1509

Depth

25-8° N., 92-5° W.,

19-3° N., 77-5° W.,

140° N.,

68-6° W.,

(m)

17 Apr. 1932

24 Apr. 1933

23 Mar. 1933

r(°C)

s(7oo)

r(°C)

5(%o)

r(°c)

5(%o)

22-94

36-16

27-32

35-99

26-08

36-38

21-90

3612

27-07

36-02

26-01

36-28

100

19-30

36-31

2508

36-04

24-97

36-68

150

16-00

3618

22-86

36-66

21-86

36-80

200

13-405

35-70,

20-35

36-67

1815

36-35

400

7-99

34-96

15-25

36-06

10-90

35-25

600

5-77

34-87

10-61

35-31

7-71

34-82

800

4-94

34-93

704

34-94

610

34-75

1000

4-54

34-93

5-14

34-90

5-18

34-84

1500

4-16

34-97

4-26

34-97

4-20

34-96

2000

4-16

34-97

4-14

34-99

408

34-96

3000

4-23

34-966

4-09

34-99

4-13

34-96

4000

—

4-20

34-97

4-25

34-96

5000

—

4-34

34-97

—

Bottom

depth

> 3000 m

5373 m

489

The question of the origin and renewal of the deep water in individual basins from
different sides has been discussed on the basis of the modern oceanographic data
collected by the "Atlantis" Expedition in the spring of 1933 and 1934 in the Carib-
bean, and by the "Mabel Taylor" Expedition in 1932 in the Gulf of Mexico. There are
only two passages through the Antilles that are important for the conditions in the
deep layers of the American Mediterranean: the Windward Passage between Cuba
and Haiti (sill depth at 1600 m), and the Anegada-Virgin Passage (sill depths at

The Three-dimensional Temperature Distribution and its Variation in Time 133

14° 16° 18° 20°C
22° 24° 26°C

500
1000
1500
2000
2500
3000
3500
4000

^J'^r—

— I — —

■"

■^ 1

.--->>--

;

.'-' 1

/
/

.^fe /

- /

-;' 4

! 1 1

•-

i
i

\ 1

IT^I

1 1

1 1 ! 1

8° 10° 12° 14° 16° 18° 20°
°C

Fig. 56. Vertical distributions of temperature for four adjacent seas.

1780-1800 m and at 1600-1620 m) between the Virgin Islands and the northern
Lesser Antilles. The Caribbean and Yucatan Basins show similar and almost constant
values for the temperature and salinity below sill depth, and it is not easy using these
values to determine the sources of the water in each basin. This was even more diffi-
cult using the older observations. However, an unequivocal solution was reached
only on the basis of the vertical oxygen distribution. Having the same potential
temperature (Yucatan Basin 3-79-3-8rC, Caribbean Basin 3-81-3-83°C) the water in
the Windward Passage contains more oxygen than that of the Anegada Passage. Since
the mean oxygen content at 2500 m (ml/1.) in the Caribbean Basin is about 5-0, in the
Yucatan Basin about 5-5-6-0 and in the Gulf of Mexico about 5-0, it follows that the
renewal by transport through the Windward Passage and that in the Caribbean Sea
is determined by that of the Anegada-Virgin Passage. The depth of the two sills can
be deduced very reliably, as shown by Dietrich, from the potential temperatures.
Earlier determinations based on the observed temperatures recorded in situ resulted
in much too large a depth. The potential temperature along a cross-section through
the Anegada-Virgin Passage is shown in Fig. 56a.

The renewal of the deep water in the Gulf of Mexico is more simple to decide.
Since the transport through the Florida Straits with a rather shallow sill depth of
about 600 m is not likely to be of great influence, the renewal must come from the
Yucatan Basin through the Yucatan Strait (sill depth 1 600 m).

{e) Vertical Temperature Distribution in Adjacent Seas at Higher Latitudes and in the
Polar Regions; Autumn and Winter Convection and Ice Formation
The basic condition for the formation of a quasi-homo-thermal state in adjacent
seas is the presence of an approximately constant sahnity at all depths below the sill

1 34 The Three-dimensional Temperature Distribution and its Variation in Time

depth. If this condition is not satisfied the convection processes in the autumn and
winter will not be able to extend to the bottom. The consequence of this limitation of
the convection to a surface layer of greater or lesser thickness is a dichothermal
temperature stratification during the warmer period of the year. There is a colder
intermediate layer situated between a warmer upper and a warmer lower layer, which
can be interpreted as the remainder of the convectional flux extending to this depth
during the cold period of the year.

1000

100
Nautical miies

Fig. 56a. Vertical distribution of the potential temperature beneath 1000 m over a vertical

section through the North American Basin, the Anegada-Virgin Passage and in to the

Caribbean Basin (according to Dietrich). Vertical enlargement by 1 : 1500.

This cold intermediate layer is typical of the whole of the open Baltic Sea during
the summer. The approximately homo-haline top layer heated by solar radiation
extends down to about 30-50 m depth; underneath a depth of 50-80 m there is
a core of relatively cold water with a temperature of 2-3°C, while still further down to
the bottom the temperature gradually rises to 4-5 °C. This cold intermediate layer re-
sults from cooling of the surface water during winter. The temperature distribution of
the top layer during this time shows an almost isothermal state due to mixing by turbu-
lence and convection, whereby at the same time the temperature at the surface may
fall to near or sfightly below the freezing point (see Fig. 51^, c; p. 116). Similar
conditions can be found in the Black Sea. For further detail see Skorzow
and NiKiTiN (1927) and especially a monograph on conditions in the Black Sea by
Neumann (1944).

During the summer the water masses in the polar waters may also show a similar
temperature distribution in the upper 100-150 m. Conditions in this layer at the end

The Three-dimensional Temperature Distribution and its Variation in Time 135

of the winter can be represented by the curve shown in Fig. 57. The winter cooling
reaches down to a depth hy,. The heating during spring and summer initially
affects only the uppermost layer and penetrates very slowly downwards to lower layers.
During the summer it reaches to a depth /?, and the vertical distribution can then be

*- Temperature

Fig. 57. Development of the vertical temperature distribution in the polar seas.

represented by the broken curve. The formation of a cold intermediate zone is clearly
shown ; it is tiot in a stationary state, but is gradually weakened by continuous heating
from above and by mixing with the warmer water masses above and below, and may
even disappear towards the end of summer to be reformed the following winter.

Table 60. The cold intermediate layer in the polar waters

Depth
(ra)

Barents Sea

"Poseidon" 15

2 Aug. 1927; 214 m

75-2° N., 260° E.

TCO

S(%o)

Cape Farewell

"Utekor" 43

9 Aug. 1930; 173 m

59-6° N., 44-0° W.

T{°C)

5(%o)

Baffin Bay

"Godthaab" 50

13 July 1928; 215 m

69-7° N., 57-4° W.

rrc)

siXo)

Labrador Current

"Marion" 1251

11 July 1931; >200m

54-6° N., 53-5" W.

TCO

5(%o)

0
10

25
50
75
100
150
175
200

+2-49
+ M9
000
-0-79
-0-79
-007
+0-26
+0-47
+0-56

30-30
3200
34-16

34-74
34-83
34-88
34-94
34-96
34-96

+0-49
+0-63
+0-98
-0-79
-0-81
+ 1-12
+2-82

32-35
32-69
.32-90
33-08
33-31
33-71
34-14

+4-10
+3-60
+0-64
-1-60
-1-56
-0-91
+0-65
+ 1-20

165 m
+2-02 34-16

213 m
+0-61 34-96

33-35
33-37
33-40
33-68
33-75
33-86
34-13
34-29

+ 3-85

+0-01
-1-19
-0-72
-0-24
+0-51

+ 1-36

32-26

3305
33-27
33-69
3400
34-21

34-47

1 36 The Three-dimensional Temperature Distribution and its Variation in Time

The cold intermediate layer is particularly pronounced and lasts longest at the
edge of pack ice and polar ice. Table 60 presents several examples. Figure 58 shows the
temperature along a longitudinal cross-section through the northern Barents Sea
74°-77° N., ]9°-38° E.) along the pack ice Hmit in August 1927 according to series
observations made by the "Poseideon" (Schulz and Wulf, 1929). From west to east
exists a layer of increasing thickness of cold winter water at a depth between 20-
1 00 m, while above this there is a layer heated by solar radiation, partly also melt
water. With distance from the ice limit this cold intermediate layer weakens and is
gradually eliminated by mixing. This cold intermediate layer forms the core of the
cold ice carrying currents around Greenland, in Baffin Bay and in the Labrador cur-
rent (Defant, 1936).

200

240

St90

74°0'N

I9°0'E

St 15 St8283 St52 53 St50 49

75°I3'NI 76°I5'N 76°32'N 77°I6'N
26°0'E 30°0'E 33°30'E 38°0' E

Fig.

58. Longitudinal temperature section in the northern Barents Sea
19°-38' E.) along the drift-ice limit (August 1927).

(74°-77° N.

The thermal structure of the Polar Sea in the layer beneath the top layer, in con-
trast to the cold intermediate layer, is determined by the deep circulation of the polar
water. In the European North Polar Basin between 250 m and 750 m underneath the
cold top layer, a relatively warm intermediate layer of water of Atlantic origin is
introduced with a temperature of about 0-5 °C (maximum of 2-0°C). Its salinity,
34-94-34-96%o, shows its Atlantic origin clearly. Underneath this layer spreads cold
deep and bottom water that reaches its lowest temperature of — 0-83°C to — 0-87°C
between 2000 m and 3000 m (WiJST, 1941, 1942). In high latitudes of the Southern
Hemisphere there is generally a similar vertical temperature distribution in all the
oceans as shown in Table 61.

Some numerical values were given previously for the annual heat exchange in ad-
jacent seas and in more or less enclosed parts of the ocean (see p. 116). The method
used for this can also be applied, as mentioned on p. 98, to the special case of the

The Three-dimensional Temperature Distribution and its Variation in Time 137

Table 61. Vertical distribution of temperature and salinity in high
latitudes of the Southern Hemisphere

Atlantic Ocean

Indian Ocean

Pacific Ocean

"WiU. Scoresby" 554

"Gauss"

"Discovery" II

Depth
(m)

5 Feb. 1931; 5143 m

26 March 1903;

13 Jan. 1931; 3098 m

3397 m

63-3° S., 17-4° W.

65-3° S., 80-5° E.

66-2° S., 71-8° W.

T(°C)

s (%„)

r(°c)

SiVoo)

TCO

s(7oo)

-0-20

33-96

-1-82

33-69

+ 1-21

33-71

-1-75

34-46

-1-5

33-69

-1-54

3406

100

-1-80

34-42

-1-6

34-35

-0-90

34-25

150

-0-35 1 34-51

-1-6

34-33

+0-30

34-43

200

+0-22 34-60

-1-6

34-30

1-43

34-58

400

0-37t

34-63

+0-05

34-48

l-65t

34-70

600

0-37

34-68

1-05

34-61

1-53

34-72

800

0-29

34-68

0-90

34-62

1-42

34-72

1000

0-20

34-67

0-75

34-63

1-25

34-72

1500

000 34-65

0-15

34-60

0-88 34-72

2000

-014

34-66

0-0

34-58

0-62

34-71

3000

n-37

'\A-f\'\

■

0-38

34-70

4000

-0-46

34-64

3397 m

-0-25

34-58

t Maximum

heat exchange at single stations. For polar stations it affords some idea not only of
the heat amounts involved in such a winter convection, but also of readiness for ice
formation at the surface of the sea which finally occurs after the temperature has been
reduced to the freezing point due to convection. These conditions can be illustrated
by an example recorded by station 888 of the "Andrey Perwoswanny" ("Murman"
Expedition) on 6 August 1903 at 71° 5' N. and 49° 0' E. in the south-eastern part
of the Barents Sea (Breitfuss, 1906). Table 62 gives the oceanographic conditions
down to a depth of 120 m, with mean values of the temperature and the density in
each layer. Layer 1 is in direct contact with the atmosphere and is exposed to all the
disturbances proceeding from it.

Table 62. "Andrey Perwoswanny" St. 888; 6 Aug. 1903 (7M° N., 49-0° E.;

126 m)

Thick-

Depth

r(°Q

5(%„)

Layer

ness

TCC)

S(%o)

Specific

(m)

volume

2-84

33-96

2-78

34-04

0-5

2-71

34-00

359

4-55

34-33

5-10

3-665

34-185

352

4-64

34-33

10-15

4-595

34-33

352

3-85

34-33

15-20

4-245

34-33

347

0-07

34-45

20-30

1-96

34-39

322

-M2

34-56

30-40

-0-525

34-5O5

300

-1-35

34-63

40-50

-1-235

34-595

291

-0-65

34-72

50-75

-1-00

34-675

285

100

-0-41

34-74

75-100

-0-53

34-73

282

120

-M3

34-81

100-120

-0-77

34-775

277

138 The Three-dimensional Temperature Distribution and its Variation in Time

At the beginning of the winter convection the temperature in this layer falls while
the salinity remains constant. When the specific volume of the first layer becomes the
same as that of the second there will be complete mixing of the two layers by con-
vection; the resultant layer will have the mean specific volume of the second layer,
given in Table 62 as 352, while the salinity will be the mean of the original salinities,
that is 3409%o. This specific volume and salinity correspond on the [r^l-diagram to

Table 63. Heat available from convection and the readiness for ice formation at

St. 888 "Andrey Perwoswanny".

Thick-

ness of

Before

nixing

After mixing

<?*

2?, +9.

Layer

the af-

M'C)

(kgcal)

5(%o)

(cm)

(kgcal)

(cm)

fected

TCO

5(%„)

spec.

TCO

S(%„)

layer

vol.

1-2

3-187

34092

352

2-85

34-092

-0-34

0-34

1-3

3-43

34- 17

351

3-45

34-17

fO-02

-0-03

—

0-31

1-4

3-65

34-23

347

3-37

34-23

-0-28

0-56

—

0-57

1-5

2-90

34-32

322

1-25

34-32

-1-65

4-95

—

5-82

1-6

0-81

34-39

300

-1-80

34-44

-2-61

10-42

0-05

0-50

16-74

7-0

1-7

-1-69

34-49

291

-1-80

34-55

-0-11

0-56

0-06

10-1

0-73

18-04

17-1

1-8

-1-53

34-61

285

-1-80

34-64

-0-27

2-00

0-03

7-2

0-53

20-57

24-3

1-9

100

-1-49

3467

282

-1-80

34-675

-0-32

3-18

0-001

0-3

002

23-78

24-6

1-10

120

-1-63

3469

277

-1-80

34-74

-0-17

2-06

0-05

17-6

1-27

28-10

42-2

a temperature of 2-85 °C. Since the mean temperature of the two layers before mixing
was 3-19°C the convection process has been accompanied by a temperature fall of
0-34 °C and the amount of heat q^^ given off from the surface will be 0-34 kg cal/cm^
by the equation on p. 96. Taking the third layer into consideration, it is now possible
to calculate the amount of heat to be removed from the two initial layers before the
third layer enters into the convection process with the two layers already mixed and
so on. Table 63 shows the final result of the mixing in each successive layer by con-
vection. However, the process proceeds in this way only until the sixth layer has been
included. After the inclusion of this layer the specific volume of all the layers cannot
reach the expected value of 300 even if the entire column of water has already been
cooled to the freezing point of salt water (— 1-8°C). At this depth the convection due
to reduction of the temperature ceases. In reality, however, after the temperature has
reached the freezing point ice begins to form at the surface and this causes an increase
in the mean salinity of the water column. From the [TlSJ-diagram it follows that the
saHnity must increase by 0-125%o for the specific volume to reach 300. From this in-
crease in salinity it is possible to calculate, using the equation on p. 96, the amount of
ice that must be formed to raise the salinity by this amount. The formation of ice
releases heat to the atmosphere; this is given by ^^ = 7-2 (c'/lOO), if e era of ice are
formed. The quantity of heat given off during the course of the convection process
down to and including the sixth layer is thus given by q^ + q^ which is 17-4 kg cal/cm^.

As long as the convection extends only to layer 5, i.e. to a depth of 30 m, there is
no readiness for formation of ice in the water found at station 888. However, when
the convection includes deeper layers it increases rapidly and when the convection
extends to the bottom it requires a layer of ice 63 cm thick.

This method presented above (Defant, 1949) is of course a little rough and not

The Three-dimensional Temperature Distribution and its Variation in Time 1 39

very precise and affords only an approximation to the time required for such con-
vection processes, but it does give a criterion of the readiness for ice formation in
polar waters. This advantage shows clearly when evaluating an oceanographic cross-
section from this aspect. Figure 59 shows such a section from the Murmansk coast
(69° N., 36° E.) in an E.N.E.-direction almost to Novaya Zembla in the southern
Barents Sea, based on measurements made by the "Murmansk" Expedition 1903.

station

883

884

885 886 887 888 889

200

Fig. 59. Section in the Barents Sea from the Murmansk coast (69° N. 36° E) north-east to

nearly Novaya Zembla.

These observations were made a little before the beginning of the convection period.
The full lines show the heat in kg cal/cm^ transmitted to the atmosphere from the
sea surface when the convection process extends to the corresponding depth. From
the Murmansk coast to about 42° E., where warm Atlantic water reaches to con-
siderable depth, conditions are uniform and there is no readiness for ice formation
even when the entire water mass down to the bottom is affected by the convection.
East of the centre of the Barents Sea towards Novaya Zembla the readiness for ice
formation increases considerably and while the amount of ice that can form is at
first not very large it reaches at the easternmost station 889 the respectable thickness
of 1 -5 m or more.

ZuBOV (1938) has developed, as it seems, a similar method for the determination
of ice potential in the ocean, without putting it into practice. He and Simpson (1954)
have again dealt with the same problem of predicting ice formation and growth and
in addition have derived new formulae for computing ice growth in terms of known
or predicted oceanographic and meteorological data. The method was used to fore-
cast the general features of the ice distribution in the Baffin Bay-Davis Strait area for
the season 1952-3. The methods for ice potential calculation have proved in practice
to give a reasonable answer for open seas, and for inshore areas where local variations
in the physical properties of the water are not large. In harbours and areas where run

140 The Three-dimensional Temperature Distribution and its Variation in Time

off is important, changes in salinity and density are too rapid to give correct forecast
values.

The forecast of ice growth based on the ice potential can only be used during the
period when ice thickness is increasing. No theory has been given which accounts
for the decreasing ice thickness during the break-up period. A mathematical theory
for this period is still needed.

The warm intermediate layer to be found at 250-750 m depth over the whole of
the North Polar Basin is of an advective nature. Its thickness depends on this, i.e.
on the strength of the oceanic circulation. It is thus not surprising that it shows strong
aperiodic variations especially in its upper boundary against the cold intermediate
layer. These variations may be as wide as 50-100 m and it is known that at the be-
ginning of this century this upper limit was at a depth of 1 50-200 m in the northern
Barents Sea and in the North Polar Sea. Since then it has risen to a depth of 75-100 m
due to the general climatic warming up of the Arctic, and in recent times the oceanic
circulation has undoubtedly increased in strength (WiJST, 1942; Weickmann, 1942).
There has been a strong increase both in the amount of ice transported into the Green-
land Sea from the central Arctic and in the transport of warmer, more saline Atlantic
water directed into the Arctic basin. Thus, since the "Fram" Expedition 1893-6, the
temperature of the warm intermediate layer of Atlantic water has risen noticeably,
as has been clearly shown by the observations of the "Sedow" Expedition 1937-40.
It is not impossible that a systematic study of these phenomena might show a close
correlation between the aperiodic variations of the boundary between the cold and
the warmer intermediate layers in the North Polar Sea and the variations in the
strength of the Atlantic oceanic circulation.

5. Temperature Distribution in Horizontal and Vertical Sections

The temperatures found at an oceanic station show the vertical temperature distri-
bution at that point, but only horizontal or vertical sections will give a two-dimen-
sional picture and thereby lead a step further towards a spatial conception of the
temperature distribution in the sea. A chart of the temperature distribution in the
Atlantic was first given by Maury in 1852 as a supplement to his charts of the winds
and currents in the Atlantic. The reliability of such horizontal temperature charts —
just as for vertical sections along fixed lines — depends on the amount of data available
and on its more or less uniform distribution over the entire section. The isotherms are
interpolated linearly between values given by observations, although it is known that
the linear interpolation does not always correspond to reality. However, the other-
wise sparse data leave too much freedom to the imagination of the analyst and the
resultant chart may soon be further from actual conditions than is tolerable.

At the present time there are several recent temperature charts available covering
the entire ocean surface. The most comprehensive presentation of surface temperature
conditions in the Atlantic has been given by Bohnecke (1936) in the "'Meteor'" Report.
The same report also gives isothermal charts for different main levels in the Atlantic ;
WiJST (1936). A selection of surface charts and such for individual depths has been
given by Schott (1935, 1942) for the Atlantic, the Indian and the Pacific Oceans.
Recent surface charts have been published by the National Hydrographic Office
in Washington (1948), World Atlas of Sea Surface Temperatures.

The Three-dimensional Temperature Distribution and its Variation in Time 141

{a) Mean Sea Surface Temperature

It seems to be unnecessary to give a detailed description of the graphical distribu-
tion of surface temperature here. Reference is made to Plate 2a and b\ this chart will
give a better conception of the actual conditions to the reader than the most accurate
description. It might be useful, however, to mention the main features of the tem-
perature distribution.

Krummel (1907) and Bohnecke (1936) have derived from the mean values for
1 0° zones the values shown in Table 64 for the mean surface temperatures of the oceans.

Table 64. Mean surface temperature of the oceans (°C)

Zone

Atlantic
Ccean

Indian
Ccean

Pacific
Ccean

Mean for
all oceans

90° N.-80° S.

16-9

170

191

17-4

The mean annual sea surface temperature of 17-4°C thus exceeds the mean annual
surface temperature of the air near the ground (land and sea) given by Hann, 14-4°C,
by a full 3°C. There is thus a considerable difference in temperature between the
hydrosphere and the atmosphere at the sea surface interface. Table 65 presents the
mean annual temperatures for the three oceans and for the entire ocean surface
separately for 10° zones.

Table 65. Mean annual sea surface temperature for 10° zones (°C)

orthern Hemisphere

Southern Hemisphere

Latitude

Mean

Atlantic

Indian

Pacific

for all

Atlantic

Indian

Pacific

for all

Ocean

oceans

Ocean

oceans

0-10°

26-6

27-9

27-2

27-3

25-2

27-4

260

26-4

10-20°

25-8

27-2

26-4

26-5

23-1

25-9

25-1

251

20-30°

24-1

261

23-4

23-7

211

22-5

21-5

21-7

30-40^

20-4

—

18-6

18-4

16-8

170

17 0

40-50°

13-4

—

100

110

8-6

8-7

11-2

9-8

50-60^^

8-7

—

5-7

1-8

1-6

5 0

60-70"

5-6

—

(-1-3)

-1-5

-1-3

-1-4

70-80°

—

-10

(-1-7)

-1-7

80-90"

—

-1-7

—

0°

-90°

°-80°

201

27-5

22-2

19-2

141 15-2

16-8

160

It can be seen from this table that the Pacific is the warmest ocean and the Atlantic
is the coldest. This is partly a consequence of the configuration of the three oceans;
the Pacific Ocean is more of a tropical ocean because three-fifths of its total surface
lie between 30° N. and 30° S. The Atlantic, on the other hand, is rather narrow
just in the tropics. The Tables also show that in the sea (as in the atmosphere) the

1 42 The Three-dimensional Temperature Distribution and its Variation in Time

thermal equator is displaced towards the north so that the temperature maximum
lies in annual average at 7° N. The large contrast between the Northern and the
Southern Hemisphere in sea surface temperature is particularly noticeable; in the
Northern Hemisphere this temperature is on the average about 2°C warmer in all
latitudes. It is especially pronounced in the Atlantic where between 50° and 60° N.
the difference is almost 7°C. This is due to the system of currents in the North Atlantic
Ocean and especially due to the general coastal configuration of the North Atlantic
which separates the water masses of the North Polar Basin, so that its cooling effect
only shows to a small extent in the North Atlantic. Analogous separation occurs in
the North Pacific. In the Southern Hemisphere, on the other hand, the three oceans
are fully exposed to the influence of the Antarctic. A further factor intensifying the
temperature differences in the Atlantic is the projection of the South American con-
tinent out to Cape San Roque in a latitude of 7° S. which deflects a considerable part
of the Southern Hemisphere tropical water across the equator into the Northern
Hemisphere.

The warmest part of the tropical ocean is a long belt with a temperature between
28 °C and 29 °C extending from the central Indian Ocean at about 60° E. through
Australian Asiatic waters to about 1 75 ° E, in the western Pacific. The western half
in the tropics is warmer than the eastern half and this circumstance is one of the most
important features of the temperature distribution in the Pacific. All the other oceano-
graphic factors are influenced in that way. In addition to this large area at a tempera-
ture above 28 °C there is also a part of the Red Sea and a small isolated area off the
south-west coast of Central America where the temperature rises above 28 °C. The
total oceanic area with a temperature higher than 28 °C amounts to 21-6 million km^
of 6% of the total ocean surface. In the Atlantic, areas with the mean annual tem-
perature above 28 °C are entirely missing.

Table 66. Area {in million square kilometres) with mean annual
temperature above 25 and above 20 "^C

In per cent

Atlantic

Indian

Pacific

Mediter-

of the total

Ocean

ranean
seas

Total area

ocean
surface

> 25°

126

> 20°

191

Table 66 shows total areas with mean annual surface temperatures above 25 and
20°C; the warm parts of the oceans aie really of enormous horizontal extent. More
than half of the entire ocean surface is warmer than 20 °C and of this 50% more
than two-thirds has a mean annual temperature above 25 °C. The oceans over much
of their surface are decidedly warm. The coldest parts of the ocean are at — 1-7°C
(close to freezing point of salt water) in the North Polar Basin and in the circum-
polar Antarctic waters.

Referring to the general distribution of the isotherms at the sea surface the following
points may be mentioned:

The Three-dimensional Temperature Distribution and its Variation in Time 143

(1) The isotherms tend to be arranged zonally, especially in higher southern latitudes
in all three oceans, where they almost parallel the latitude circules. This is due to the
homogeneous climatic conditions over this almost exclusively oceanic area.

(2) The major equatorial ocean currents to a large extent run from east to west.
At east coasts of the continents they diverge and the isotherms do the same. The
western sides of the oceans are thus appreciably warmer than the eastern sides. These
differences are particularly pronounced in the Atlantic; here in temperate and higher
latitudes this difference between east and west is actually reversed, and from about
35° N. the east is appreciably warmer than the west. However, this phenomenon
does not occur in the Southern Hemisphere. Again, the major current system at the
sea surface can be considered to be the cause of different behaviour of both hemi-
spheres. The horizontal advection of water with a different temperature produces
almost stationary contrasts in temperature between the eastern and western side of
the ocean. In addition the distribution of land and sea and in some regions local
oceanographic-meteorological phenomena, such as upwelling water, and piling up
("Anstau"), influence the temperature distribution.

(3) There is another phenomenon apparent on the chart which is not clearly shown
in the Southern Hemisphere because of the sparsity of the observations, although it
has long been recognized in the Northern Hemisphere. This is the uneven, stepwise
change in temperature towards higher latitudes. Already Fig. 50 (see p. 1 13)shows clearly
this phenomenon, as it appears in the Atlantic. In both the Northern and the Southern
Hemisphere there is an increase in the meridional temperature gradient in the zone
between 40° and 50° which, during the year, is displaced towards and away from the
poles following the movements of the sun. The concentration of the isotherms into a
narrow belt between the Gulf Stream and the Labrador Current and between the
Atlantic water and the Greenland Current is quite obvious. This boundary is called, in
analogy with the atmospheric polar front, the "oceanic polar front" which indicates
the position of the Arctic convergence where the two different types of water are
brought into close contact. Its southern continuation along the east coast of North
America has long been known as the "cold wall". This discontinuity appears in the
chart of mean values because the aperiodic displacements of the ocean currents are
confirmed within narrow limits. Accurate information about this sharp discon-
tinuity has only been obtained from numerous thermographic recordings made by ship-
ping across the whole system of currents off the east coast of North America (Church,
1937; Spillhaus, 1940). Figure 60 shows the most important of the results obtained by
analysis of these recordings. The coastal water with a slowly increasing temperature
eastwards borders the warm belt of water in the Gulf Stream which is barely 50 km
wide. Towards the east the Gulf Stream is separated almost as sharply by a rapid
fall of temperature from the water of the Sargasso Sea, where the temperature rises
again slowly towards the east and south-east.

The "band" character of the Gulf Stream does not show very clearly in the hori-
zontal temperature charts, since the temperature is recorded at one or two degree
squares which completely blurs this phenomenon, and the strong aperiodic dis-
placements of the discontinuity along the right-hand side of the band (looking down-
stream) contribute to this blurring when mean values are taken. The observations are
also not strictly synoptic but are only obtained with differing time.

144 The Three-dimensional Temperature Distribution and its Variation in Time

° 20

^ 16

o
i>

_^^^- —

Gulf
She
Co

streonr
f woter
istol wc

' /

Sargasso- Sec

2°

0 200 400 600 800 1000 1200

Sea miles towards SE

Fig. 60. Surface temperature distribution in the western North Atlantic (in the area of the
Gulf Stream) from repeated temperature recordings made along shipping routes (according

to Church).

In the western part of the North Pacific there is also a similar phenomenon at the
boundary between the warm Kuroshio and the cold Oyashio where arctic water and
subtropical water come advectively in close contact.

Due to the lack of data it was for a long time impossible to determine the position
of this discontinuity in the circumpolar water in the Southern Hemisphere. Meinardus
(1923) first showed its presence from observations made in the southern Indian Ocean.
Its position in the Atlantic was deduced later from the current charts and it was
recognized as the line of covergence between the oceanic west wind drift and the Ant-
arctic water (Defant, 1928). It runs from about 48° W. to well out into the Indian
Ocean (80° E.) between latitudes of 50° and 48° S. and then gradually turns south-
wards to about 62° S. at Drake's Passage.

(4) A second temperature discontinuity which is sometimes more sharply marked,
though it can still only be detected on continuous recordings, lies where the sub-
tropical water meets the subarctic water of the oceanic west wind drift {subtropical
convergence). The frontal discontinuity in the region of the subtropical convergence
shows large local meridional displacements and is therefore completely smoothed in
mean temperature charts. Figure 61 shows two thermograph recordings given by
Deacon (1938) that were taken on passing through the subtropical convergence and the
Antarctic convergence {oceanic polar front). They show clearly the character of frontal
discontinuity of this dynamically important phenomenon.

(5) A useful aid in comparing temperature conditions in the oceans, especially in
a zonal direction, are charts with lines of equal deviation from the normal value charac-
teristic for each latitude. Such isoanomalic charts show which parts of the ocean are
cold and which are warm relative to a normal latitude. In the Atlantic the heat surplus
from the Gulf of Mexico across the North Atlantic to the Norwegian Sea as far as

The Three-dimensional Temperature Distribution and its Variation in Time 145

Spitzbergen is particularly noticeable. This warm zone is associated with the Gulf
Stream. There are negative anomalies showing the advection of polar water in the
east Greenland Sea and the Labrador Sea down to Newfoundland. The Moroccan
and the south-west African areas of upwelling water also show negative anomalies,
and the eastern side of the Atlantic south of 35° N. is colder than the west side. A
similar phenomenon also appears in the South Atlantic. The Pacific generally shows
a similar subidi vision, with the western half decidedly warmer and the eastern half
too cold.

12 IS 20 24 4

12 16 20 24" 4

20 24" 4. 8 12" 16 20 24 4 6 12

\ \ v \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

\\\\\ rl^^fc^pff^^^ri^^^

Fig. 61. Thermograph recordings made passing through the subtropical and antarctic con-
vergences (according to Deacon).

{b) Horizontal Temperature Distribution at Different Depths and Vertical Temperature
Sections

The horizontal temperature distribution remains similar to that at the surface down
to a depth of at the most 50-75 m and then changes rapidly. It was already shown in
Table 48 that the thickness of the top layer {disturbance layer) is least in the equa-
torial areas and this is where the cold water masses of the subtroposphere come
closest to the surface. It is thus to be expected that horizontal temperature charts,
even for shallow depths, will show a band of cold water embedded between the warm-
water masses of the subtropics which becomes greater in width with increasing depths.
This can be seen on horizontal temperature charts at 200 m intervals both for the
Atlantic and also on charts for the other oceans. The subtropical warm-water areas
of both hemispheres are thus separated by a cooler equatorial zone almost 30° wide
and are limited polewards by two cold-water areas in higher latitudes. In the layers
between 400 and 800 m the highest temperatures are always found on the western
side of the oceans, particularly in the Atlantic. This is a dynamic consequence of the
stationary distribution of the currents at these depths.

The chart for a depth of 800 m shows already the asymmetry typical for the tem-
perature distribution in the deep layer of the Atlantic, which is due to the cold sub-
antarctic intermediate current in the south and to the influence of the Gulf Stream
and the inflow of Mediterranean warm water in the north. This asymmetry domi-
nates the temperature distribution down to depths of more than 3000 m. The influence

1 46 The Three-dimensional Temperature Distribution and its Variation in Time

The Three-dimensional Temperature Distribution and its Variation in Time 147

3
O

Ilk _>i

. 60

en
>^

'5 "o
c "-^

O <4_

w o

.3
•a

a
o

148 The Three-dimensional Temperature Distribution and its Variation in Time

of the Gulf Stream extends down to about 3000 m and that of the Mediterranean
water down to 2000 m, so that there is always a considerable heat surplus even at
these great depths in the North Atlantic. Below 3000 m the cold Antarctic bottom
water first appears and at deeper levels spreads northward with slowly increasing
temperature. There are insufficient systematic data available for the Indian and
Pacific Oceans below 2000 m to allow any reasonably accurate description of the
horizontal temperature distribution in the deeper layers.

The importance of horizontal charts of the distribution of temperature and other
oceanographic factors, as a geographic aid to the comprehension of the distribution
of these factors throughout the ocean, has in the past been somewhat overestimated.
Oceanic processes never, or very rarely, occur along horizontal planes or are quasi-
horizontally arranged. Because the three-dimensional field of oceanic elements is
arbitrarily intersected by horizontal planes, connected phenomena will therefore be
cut by such planes. They are thus, for example, quite insufficient for following water
movements in the depths of the oceans. The same is equally true for the study of
atmospheric phenomena. Before these were deduced in other ways it was difficult to
interpret the arrangement of the isotherms in horizontal sections. In all cases vertical
cross-sections must also be used to clarify the three-dimensional field of any oceano-
graphic element.

Vertical temperature sections can be taken in any direction and thus can give a far
better idea of the thermal stratification of a water mass than a horizontal chart. It
is, of course, best and most convenient to take the vertical section either along the
axis of major spreading of the water mass in the ocean concerned or across it.

At the present time there are several such longitudinal or transverse vertical sections
(relative to the direction of flow) for all three oceans, showing temperature, salinity
and in part also the oxygen content. Those for the Indian Ocean (Moller, 1929;
Clowes and Deacon, 1935) and for the Pacific Ocean (Wust, 1929; Sverdrup, 1942,
1945; ScHOTT, 1942) are less accurate because of the smaller number of stations than
those for the Atlantic Ocean (WiJST, Defant, 1936). It is neither possible nor appro-
priate to describe and interpret these vertical sections individually. An interpretation
can only suitably be given in conjunction with the phenomena of the oceanic circula-
tion in the deeper layers. Figure 62 shows, as an example, a longitudinal section along
the western side of the Atlantic giving temperatures and salinities (after WiJST, 1928).
This runs from 75° S. near the area of formation of the Antarctic bottom water,
through the Weddell Sea and the South Antilles Sea, along the western side of the
West Atlantic Trough to the Newfoundland Banks through the Labrador Basin to
the Davis Ridge. There is a vertical distortion of the section by a factor 1 : 1300. This
section is quite typical of all sections through the Atlantic Ocean and shows the im-
portant characteristics of the meridional vertical temperature distribution: the two
large warm-water accumulations in the subtropical troposphere of both hemispheres,
the approach of the cold-water mass in the equatorial subtroposphere towards the
surface, the concentration of the isotherms at the polar limits of the troposphere
between 40° and 50° S. and 45°-55° N., and the oceanic polar fronts. This western
section also shows at about 1000 m an intrusion of colder water from 55° S. towards
the north as a tongue-shaped bulge on the isotherms which is visible even across the
equator. In a central section this is only weakly developed, in an eastern section it is

The Three-dimensional Temperature Distribution and its Variation in Time 149

not visible at all. It is caused by the intrusion of subantarctic intermediate water and
represents the same phenomenon as the isothermal layer or actual inversion in the
vertical distribution which was mentioned previously (see p. 123). South of 55° S.
the oceanic space all around the Antarctic is filled down to the greatest depths with
cold Antarctic water. The isotherms here steeply descend from the surface to 2500-
3000 m, clearly showing the extension of this cold-water type northward along the
deep basins that open to the south.

This, like all other longitudinal sections, shows the considerable asymmetry in the
temperature distribution of the oceans. As previously mentioned this asymmetry
is caused by topographic conditions of the Atlantic, which allow only a spreading of
the cold heavy Antarctic bottom water towards the north. This is, of course, also the
case in the Indian Ocean but not entirely so in the Pacific where, although only to a
small extent, there is an Arctic component from the Okhotsk Sea to be taken into
account. The meridional temperature contrast between high-southern and high-
northern latitudes, which is especially well shown in the Atlantic and can also be seen
in the Pacific Ocean, is the main cause of the deep-sea circulation of these oceans and
also gives rise to their asymmetry relative to the equator.

The question of the origin and the spreading of the lowermost layer of bottom
water in the oceans was raised at a very early stage in the development of oceano-
graphy— much earlier than the problems dealing with the oceanic circulation of the
middle layers. This was due to the existence of a greater amount of data for the
bottom layer than for the middle and deep layers, since bottom temperatures were
measured from cable-laying ships as well as from research vessels. The low tempera-
tures found in the bottom layers clearly indicated at an early stage a polar origin of
the bottom water and formed the main basis for the assumption of a deep-sea circula-
tion. An historical account of the exploration of the nature of the bottom water has
been made by WiJST (1936), who has also given a description and comparison of the
movements of the bottom water spreading out into the three oceans based on a critical
inspection of all the available data (Wust, 1938). Plate 4 gives a chart of bottom
temperatures on the deep-sea basins. The course of the isotherms is much more cer-
tain in the Atlantic than in the other incompletely explored oceans. The temperatures
given are potential temperatures in order to give a clear picture of the spreading of
bottom water influenced by the relatively large irregularities of the bottom topography.
Table 67 gives mean values for 10° latitude zones in the three oceans and for the total
ocean. In general, there is a continuous rise in the bottom temperature to be seen from
high southern latitudes across the equator as far as to temperate northern latitudes.

The maximum temperature that can be taken as the boundary between Arctic and
Antarctic influences at the bottom is situated rather asymmetrically at 40° N. in
the Atlantic and at 30° N. in the Pacific. In almost all latitudes the coldest bottom
water is found in the Indian Ocean. The coldest water is in the deepest depressions
in the Atlantic South Polar Basin; the cold pole with — 0-92°C lies at the western
edge of the Weddell Sea, where according to Brennecke (1921) and Deacon (1937)
that thermo-haline stratification in the autumn and early winter exists, which per-
mits the ice-cold shelf water to sink by convection along the continental slope down

1 50 The Three-dimensional Temperature Distribution and its Variation in Time

to the ocean bottom. From here this cold heavy water spreads out in general towards
the east within the Antarctic circumpolar Ocean to form the source of the meridional
northward outflow along the deep-sea troughs of the Ocean. It is still uncertain
whether there are other regions of bottom-water formation in the Antarctic, but
that in the Weddell Sea is in any case the most important and the most intense one.
In each ocean the Antarctic bottom water spreads out both in zonal and meridional
direction according to the bottom topography. There are seven cold streams of bottom
water spreading out along the seven major longitudinal troughs of the oceans towards
the north. These are listed in Table 68.

Table 67. Mean zonal distribution of bottom potential temperature
(°C) in the deep sea (> 4000 m); mean for each latitude circle,

(After WiJST 1938)

Latitude

Atlantic

Indian

Pacific

All oceans

Ocean

S. 70"

-0-71

-015*

-0-43*

-0-87*

"0-54*

006

-0-42

-0-33

0-25

0-49

012

017

0-36

0-67

0-44

100

0-53

0-84

0-76

104

0-61

103

0-90

S. 10"

119

0-86

103

101

1-32

0-93

106

107

N. 10

1-66

M6t

108

1-20

1-89

— .

108

1-32

1-83

—

llOf

l-33t

l-95t

—

100

1-32

N. 50°

1-81

—

106

1-22

Strongest meridional

difference

2-82

1-70

125

1-76

* Minimum; f Maximum

Table 68. Initial temperatures and northward extent of the cold
Antarctic bottom water

Initial

temp. (X)

at 55° S

Northern extent of cold water

Deep-sea Trough

tongue (potential temp. 1 •0°C)

To lat.

To cross ridge

1 . West Atlantic

-0-8

8°N.

Para Rise

2. East Atlantic

-0-7

22° S.

Whalefish Ridge

3. West Indian

-0-6

10° N.

Carlsberg Ridge

4. East Indian

-0-3

5°N.

—

5. Western Pacific

(Tasman Basin)

0-2

24' S.

Coral Rise

6. Central Pacific

0-4

25° N.

Hawaii Rise (?)

7. Eastern Pacific

(South Polar Basin)

37 S.

Eastern Rise (?)

The Three-dimensional Temperature Distribution and its Variation in Time 151

The distance to which each of these streams extend in each meridionally-oriented
trough is very largely dependent :

(1) on the morphological form of the trough, on whether there are deep passages
through cross-ridges or whether the stream can flow over any rises, and

(2) on the kind of water mass spreading above the cold bottom water towards the
equator. It combines and interchanges with this and shows much stronger conserva-
tism in its character of Antarctic water the lesser the influence of the water above.
The most extended is the central Pacific cold stream which, due to the favourable
topography and partly also because of the absence of deep warm currents in the North
Pacific, reaches as far as 25° N. Also, in the Indian Ocean, the cold-water currents
on both sides of the central ridge extend almost to the northern limit of the ocean.
The most impressive one of these streams is, however, the west Atlantic cold water
spreading where the Antarctic water penetrates through gaps from deep-sea basin to
deep-sea basin as far as the Para Rise at 8° N., and finally warms up by mixing with
the relatively warm North Atlantic deep water and flows into the North American
Basin. In the East Atlantic Trough the Whalefish Ridge completely prevents further
extension north and there is therefore a large difference in the temperature of the bot-
tom water on the north and south sides of this cross-barrier. The bottom layers of
the Atlantic Eastern Trough north of the Whalefish Ridge are formed by colder West
Atlantic bottom water flowing in through deep gaps in the central parts of the Middle
Atlantic Ridge at 0° latitude (Romanche Deep) and at 10° N. There are cross-rises
also in the eastern and western deep-sea troughs of the Pacific that prevent the north-
ward extension of Antarctic water beyond 22° and 37° S., respectively.

There is very little bottom water of Arctic origin. The most productive source is
probably the outflow from the Okhotsk Sea which extends southwards as a cold
stream, with an initial temperature of less than 0-6 °C about 15° N. In the Atlantic
deep-sea troughs there are indications of bottom water at less than 1-8° between 53°
and 45° N. which is probably of subarctic origin.

A detailed investigation of the horizontal spreading of the Antarctic bottom water
in the Atlantic has been made by WiJST (1936). Figure 63 shows the potential tempera-
ture along a quasi-meridional section through the Western and Eastern Troughs below
3000 m. In the western section the bottom water is separated from the water mass
above by a marked discontinuity in the vertical temperature (and salinity) distribution.
It descends from south to north with a gradient of about 20 m in 100 km and follows
the bottom topography closely. Such influences on the temperature (and salinity)
are recognized as far north as 40° N., 16,500 km away from the origin of the stream
at the rim of the South Polar Basin.

The eastern quasi-meridional section is rather different. The barrier due to the
Whalefish Ridge shows even more prominently here and the eff"ect of the local inflow
of Antarctic- West Atlantic water through the Romanche Trench is also clearly visible.
From here and from the saddle at about 10° N. the bottom water spreads north and
south in the eastern Atlantic Basin. The increase in temperature and salinity along
the core of spreading of the relatively shallow bottom water is due to mixing processes
with the warmer North Atlantic deep water above, comparatively of larger vertical
extent. The distribution of temperature and salinity in the bottom water can be re-
garded as stationary and this can only happen when advection and mixing are in

1 52 The Three-dimensional Temperature Distribution and its Variation in Time

l-K-

\:M.

.i C « (3

I g § s

^ Xi
C bO

Si o

B ■"

^W o

:3 W) ^

.S .2 3

2 o » § >^

^o 3 S

CM ^ O

.2 O

S
o

13 o

C XI '

■5 '->

3 '-n
■>-• o

o c

; J «
o

£ o

X MH 60

i: o c

^ *- t? w
•^ I ^ '"

6--3 SsS

LU 'mdSQ

'Mtdea

The Three-dimensional Temperature Distribution and its Variation in Time 153

balance. From the distribution of these factors the ratio of the vertical exchange ^2 to
the velocity u of the spreading can be calculated (Defant, 1936). The value of A ^lu
is between 2 and 3 over the transverse rises and between 5 and 6 in the troughs, with
a maximum value of 10. Because this ratio as a first approximation is proportional
to the Prandtl mixing length (see Chap. XII I) and this length is more suited for the charac-
terization of a turbulent flow than A 2 the above result therefore means that the mixing
length is greater in the troughs than over the rises. In the core of this flow for a
narrowing of the gap and corresponding increase in the velocity the mixing is somewhat
reduced (more laminar flow), while in basins, on the other hand, the contrary occurs
(velocity-decrease, stronger mixing).

6. Mean Vertically Integrated Temperature for Individual Oceans in Zonal Rings

Calculations of mean temperatures of parts of the sea, or of particular zones of
latitude or for the total ocean, are of course only of statistical value. Krummel (1907)
determined the values of some of these mean temperatures on the basis of the hori-
zontal charts then available; Table 69. The mean temperature of the total ocean of
3-8 °C appears very low especially compared with the surface value of 17-4°C. The
decisive factor is the very large water masses of the oceanic stratosphere and the com-
paratively shallow oceanic troposphere. The mean values for 10° latitude zones show
again the marked decrease of about 5°C between the equator and higher latitudes, but
the differences between 40° N, and 30° S. remain, in general, small. This is also true
for differences in the values for the three oceans.

Table 69. Mean vertical integrated temperatures °C for different oceans and the

total ocean
(According to Krummel 1907)

Northern Hemisphere

Southern Hemispheie

Zone of
latitude

Atlantic

Indian

Pacific

All

Atlantic

Indian

Pacific

All

Ocean

oceans

Ocean

oceans

0-10°

5-8

4-5

4.9

4.4

5-2

4-6

4-7

10-20°

5-1

7-4

4-8

4-2

4-8

4-7

20-30°

5-8

10-3

3-8

4-7

4-8

4-5

4-6

30-40°

— •

4-5

3-7

4-2

4-1

40-50°

—

2-4

3-2

2-1

2-6

2-8

50-60°

3-8

—

2-3

•2-8

0-6

0-8

1-4

60-70°

4.4

—

2-2

-0-2

0-4

70-80°

—

(-0-6)

-0-2

0-3

80-90°

—

(-0-9)

—

0-90°

(resp. 80°)

5-3s

6-5,

3-6e

4-3,

7.0

3-4,

3-72

3-4,

90° N.-

80° S.

40,

3-82

3-73

—

On the whole, the mean temperature of 3-8 °C for the entire ocean makes a rather
cold environment for the living organisms in it, however, they are mainly concen-
trated in the upper warmer layers.

Chapter IV

The Salinity of the Ocean, its Variation
in Oceanic Space and in Time

1. Periodic and Aperiodic Variations of Salinity *

If tidal effects are disregarded the most obvious periodic changes in salinity to be
taken into account are the diurnal and annual variations. There is little data on daily
variations. The diurnal variation of evaporation must give rise to a similar change in
the salinity but it can have only little signification. Apart from the small diurnal
variation in evaporation, the variations in salinity will be further diminished by the
vertical convection set up immediately in the homogeneous top layer by increased
salinity at the surface. The effect of an increase in salinity by a high evaporation rate
will thus spread very rapidly over a large water mass and will scarcely be detectable.

The true salinity variation uninfluenced by other factors can only be shown by ob-
servations made at an oceanographic anchor station, and in this case also all stations
that showed any appreciable vertical salinity gradient should be left out of account.
At such stations the vertical displacements of water by the tides cause variations in
salinity with a tidal period which are usually several times greater than the normal
diurnal variations. A small diurnal variation can only be clearly shown in an almost
completely homo-haline top layer. Five "Meteor" anchor stations between 21° S.
and 4° N. gave the mean diurnal variation shown in Table 70.

The second column of the table shows the diurnal salinity variation as hourly
values taken over three days at the "Altair" anchor station (44-5° N., 34° W.); see
Fig. 64. The range is very small and amounts to less than half of 1/100 part %o; there
is a broad flat minimum during night time until sunrise after which the salinity rises,
slowly at first and then rapidly, to a pronounced maximum in the late afternoon and
falls off just as rapidly to the night values. Physically the process can be regarded as
the effect of a positive transient source of salt at the surface, the surface amplitude
of the effect being somewhat modified by vertical exchange with the layers under-
neath. The variation proceeds so regularly that despite its small amplitude it deserves
more attention than it has hitherto received. Visser (1928) deduced a value for the
mean diurnal variation of the surface salinity by analysis of the observations of the
"William Snellius" Expedition; this is similar to that found in the Atlantic: minimum
at 04.00 h, maximum at about 17.00 h; but the amplitude was almost twice as large
probably due to climatic conditions in the area.

Knowledge of the annual salinity variation is also rather meagre. Bohnecke
(1936) has prepared charts showing surface salinities for each month in the North
Atlantic and seasonal means of salinity for the total Atlantic which allow the annual

154

Salinity of the Ocean, its Variation in Oceanic Space and in Time

155

salinity variations to be found; these are supplemented by a chart showing mean
annual amplitudes. Over the major part of the open ocean surface away from coastal
areas the annual range in salinity in middle latitudes is less than 0-5%o, usually less
than 0-25%o. A zone with more than 0-5%o and a core with more than \%o and oc-
casionally over l-5%o extends right across the Atlantic from South America to Africa
between 5° and 15° N. and includes the area of the equatorial counter current. There
is a further region with values greater than 0-5%o and several cores about l%o in the
Gulf Stream region until the south-east of the Newfoundland banks. Otherv^dse the
maxima of annual variation are found in coastal areas especially off the mouths of the
larger rivers (Amazon, La Plata, and the inner part of the Gulf of Guinea) with large
seasonal variations in fresh-water inflow or in polar areas with seasonal melting of the

Table 70. Diurnal salinity variation

Five "Meteor"

"Altair" anchor-

Time

stations

station

(hours)

2-l = S.-4°N.

44-5"N.-34-0=W.;
3 days

35-468

35-800*

35-466

35-866

35-464*

35-887

35-464

35-876

35-466

35-882

35-470

35-889

35-480

35-885

35-490

35-893

35-540t

35-913t

35-486

35-900

35-474

35-883

35-466

35-879

Range

0-042

0-040

Minimum; f Maximum

Fig. 64. Diurnal salinity variation. Above: the mean of five "Meteor" anchor stations

between 21 ° S. and 4° N. in the Atlantic Ocean. Below: the mean for three days at the Altair

anchor station (44-5° N., 34" W.).

156

Salinity of the Ocean, its Variation in Oceanic Space and in Time

ice (especially around Greenland, Tierra del Fuego and similar places). A special
investigation of the annual salinity variation in the open North Atlantic has been
made by Smed (1943).

Neumann (1938) has made a detailed investigation of the annual temperature and
salinity variations over twelve five-degree squares for part of the Gulf Stream region
between Newfoundland and about 25° W. (north and north-west of the Azores).
These variations are presented graphically in detail in Fig. 65. It shows a rapid decrease

Fig. 65. Annual salinity variations in the North Atlantic between the Newfoundland Banks
and the Azores (according to Neumann).

in the annual amplitude and a displacement of the maximum on moving from the
west to the east and south-east away from the Newfoundland Banks, where the large
annual change in salinity is due in the first place to seasonal changes in the inflow "of
salt with the Labrador Current. This area is the starting point of an annual disturb-
ance that spreads out to the east and south-east and gradually diminishes in intensity
due to mixing. This phenomenon can be treated theoretically! and comparison with ob-

t The differential equation governing the process requires that the local change dsjdt of salinity
with time and the change by horizontal salinity advection u(8sldx) should be exactly balanced by the
change in salinity due to mixing {Aylp)(8Hldy^) so that

Ss , 8s A^ 8^s

^r + « — = — ^ — s-
8t 8x p 8y^-

The boundary condition for a linear increase in salinity from y = —m to y = +m on which is
superimposed a periodic disturbance at j: = 0 with a maximum amphtude at the zero point and
vanishing at >> = ±'n may be formulated as

S-^o = ^ + ^y + C cos -^ cos — .
2m T

Then a general solution can be given in the form

s = M + Ny + Cexp \~^'LA^]
L 4i>rpiii

This solution gives a salinity distribution that varies with time in the region from +m to —m as a
function of distance and time. The intensity of the disturbance decreases in the direction of flow
according to a power of e-function.

nV
COS — COS

?-H'-l)l

Salinity of the Ocean, its Variation in Oceanic Space and in Time

157

served values leads to a maximum lateral exchange coefficient of 4-9 x 10^ gcm-^ sec^^
which in view of the intense mixing in the Gulf Stream is of an order of magnitude
in good agreement with this coefficient (see p. 105).

From the extensive data available for the Australian-Asiatic Mediterranean (largely
from the "William Snellius" Expedition) Visser (1928) has determined the annual
temperature and salinity variations and has discussed them in detail. The rather large
annual variations here (more than 2-5%o) are also mainly produced by advection.
Table 70a gives, as an example, some values for the eastern Java Sea. While the tem-
perature shows the equatorial double wave with maxima in April and December and
minima in January and August, the salinity shows only a single main maximum in
September and single minimum in May. These phenomena are due to the monsoon
change and the associated changes in advection. During the east monsoon cold sahne
water flows in from the east (May to August) and the salinity rises ; it remains almost
constant during the monsoon change (September to November) and falls from De-
cember to February, while the west monsoon carries water of lower sahnity in from
western Java Sea.

Table 70a. Annual temperature and salinity variations in the eastern Java Sea

Jan.

Feb.

Mar. Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Annual
range

Temp. (°C) 26° plus:
Salinity(%„) 31 plus:

1-88*
0-71

1-96
0-84

2-25
1-38

2-66t
(0-88)

2-41
0-39*

1-79
(1-24)

105
210

0-74*
2-73

0-95
2-98t

1-99
2-57

2-44
2-56

2-47 1
(1-64)

1-92
2-59

* Minimum; t Maxi

mum;

()Ap

proxima

te valu

In the interval between the monsoons from March to May the changes are only
small. It is obvious that here also the advection of water masses of different sahnities
is the principal factor involved.

In the Polar regions the annual salinity and temperature variations may be due not
only to the effects of advection but also to ice formation and melting thereby producing
large amplitudes. The annual salinity variation may be increased to as much as 25%o
or more, but this occurs only in a very thin top layer; the layers underneath show only
a small annual variation with a maximum in winter and a minimum in summer. This
small annual variation can be regarded as a consequence of ice formation. Table 71
shows, as an example, conditions in the homogeneous top layer of the east Siberian
Sea from November 1922 to October 1923.

SvERDRUP (1929) pointed out that between February and the end of May there was
an increase of 0-47%o in the salinity of the layer below the top layer. If this is assumed
to be due to ice formation, and the ice formed is assumed to have a salinity of 5%o,
then the increase observed corresponds to an ice layer 67 cm thick which is in agree-
ment with actual measurement of ice thickness. The salinity decrease between May
and August is about 0-55%o, corresponding to the melting of 87 cm of ice which is
also in agreement with the observed values.

Footnote continued from opposite page

Knowing the amplitude of the variation in the region of the flow the quantity Ayj pu can be calcu-
lated and knowing p and u a numerical value of the lateral exchange coefficient Ay can be found
(see p. 106. et seq.).

158

Salinity of the Ocean, its Variation in Oceanic Space and in Time

Table 71. Monthly mean values for T and S in the homogeneous top layer in the East
Sebirian Sea, Nov. 1922-Oct. 1923

Depth
(m)

1922

Nov.

Dec.

1923

Jan.

Feb.

Mar.

Apr.

Temp. °C

Salinity (%„)

0
10-30

-1-63*
-I-6I2

29-45
29-50

-1-61
-1-62,

29-56
29-50

-1-60
-1-593

28-99
29-23

-1-57
-l-59o

29-21
29-20

-1-58
-I-6O0

29-28
29-36

-1-57
-I-6O5

29-49
29-46

Depth
(m)

1923

May

June

July

Aug.

Sept.

Oct.

range

Temp. °C
Salinity (%„)

0
10-30

-1-58
-l-62o*

29-67t
29-67

-0-98
-1-587

29-25
29-71t

0-80t
-1-552

24-70
29-61

0-47
-l-48et

23-58*
29-14

-0-21
-1-498

24-79
28-74

-0-35
-1-524

27-57
28-56

2-43
0-134

6-09
1-15

* Minimum; j Maximum

The annual variation in the surface salinity in an adjacent sea depends very largely
on whether it has a humid climate with a large inflow of fresh water from rivers and
from precipitation, or whether it is in an arid climate with little fresh-water gain but
with a high evaporation rate. The latter type of adjacent seas with high salinities show
only a small annual variation, since the evaporation has very little effect on the surface
salinity; in the first type, on the other hand, the annual range may reach relatively large
values. The annual surface salinity variation at the Adlergrund hght-ship in the south-
western part of the Baltic is presented as an example (Neumann, 1938) (Mean monthly
values for the period 1926-35) in Table 72.

Table 72. Annual surface salinity variation at the Adlergrund light-ship (Baltic Sea)
and total fresh-water inflow into the Baltic

1 Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Mean

Range

Salinity (%„) 7-51

Variation
Inflow

(km'/ 231
month)

7-52t
26-8

7-50
34-4

7-45
59-8

7-39

84-2t

7-30
71-0

7-31*
480

7-31
41-2

7-32
30-2

7-36
26-4

7-41
23-5

7-48
22-8*

7-41
40-09

0-21
61-4

• Minimum; t Maximum

These values follow almost exactly a pure sine curve
S = 7-41 + 0-103 sin | ^ t + 66-8 | + 0-006

l-^ t + 66-8 j + 0-006 sin I -

/ + 15-4'

with a maximum in January-February and a minimum in July-August. The most
important factor affecting the amount of salinity in the Baltic is the inflow of fresh

Salinity of the Ocean, its Variation in Oceanic Space and in Time

159

water from rivers and from precipitation. There is therefore a very close correlation
between the two phenomena if one applies a readily explicable phase difference of
about two months. Gehrke (1910) has already pointed out that this phase difference
becomes smaller and smaller approaching the coast from the open sea. Similar con-
ditions are found in the eastern part of the Baltic (Granquist, 1938).

Of the occasional disturbances in the surface salinity occurring only for very short
time due to the influence of external agencies, probably the most interesting is that
caused by precipitation. It is to be expected that heavy precipitation of long duration
will reduce the salinity. However, this reduction, first affecting the surface, will extend
when precipitation continues down to deeper and deeper layers beneath the surface
due to turbulent mixing. After the cessation of the precipitation there is a continued
equalization of the sahnity change by these turbulent processes that gradually ehmi-
nates the disturbances. Some data are found in the literature (Krummel 1907) on
the relationship between precipitation and simultaneous and subsequent decreases in
salinity, but these observations have been made from moving vessels and therefore
do not permit an unequivocal quantitative determination of the effect of dilution
by precipitation. Neumann (1940) has given some more recent results on the determina-
tion of salinity before, during, and after rain. Figure 66 shows three sets of observation
made by the research vessel "Carimare" of the surface salinity given as deviations
(l/100%o) of the value at the time of the rain. In total agreement with each other all
three cases show minimum salinity at the time and at the end of the rain. After the
rain the salinity rose, at first rapidly and then more and more slowly to the value

-8 -7 -6 -5 -h -3 -2 -1 0 +1 +2 +3 *h +5 +6 +7 +8 +9 +10

%(?>oo '

■ 1

— 1 —

130

■^^,^^^

120

100
90

1
1
1
1

1
1

60
50

1
1

kO
30
20
10

•

. —

"^x 1

1
1

1
1
1

1 ^

1..

1
1

1 f

"■ T

" "•

Fig. 66. Changes in surface salinity due to precipitation (according to the observations of the
research vessel "Carimare"). The zero point on the abscissa corresponds to: in case 1 (6-7
June, 1938 at 1 7.00 h) ; in case 2 (8 June, 1938 at 04.00 h) ; in case 3(16 June, 1938 at 02.00 h).

160 Salinity of the Ocean, its Variation in Oceanic Space and in Time

present before the disturbance produced by rainfall, so that 2-3 h after the rain had
ceased the salinity values did not differ from the value before the rain by more than
0-05-0-10%o.

A quantitative treatment of these processes has been given by Defant and Ertel
(1939). The rain v/ater falling on the surface can be regarded physically as a salinity
sink at the surface (z = 0) that consumes a quantity of salt —S per unit time and unit
area; this corresponds to an intensity in the salinity flux —A^ids/dz) immediately at
the sea surface z = 0 (^4^ is the vertical exchange coefficient, s is the amount of salt
in unit mass, z is counted positive downwards). This reduction in salinity extends
downwards into deeper layers by mixing during the precipitation period according to
the exchange equation

ds A 8^s
'dt^ ~p 8z^'

At the start of precipitation (/ = 0) the salinity should be uniform {s = ^o)- ^ will be
dependent on the intensity of precipitation and on the time t and therefore for a dura-
tion T of precipitation

2J(t) > 0 for 0 ^ t ^ T

while at the end of precipitation

2:(t) = 0 for t ^ T.

At large depths the disturbance will vanish so that for z = oo and for any time s =s*.
Solution of the problem for the given boundary conditions will give a complete
answer for the entire process not only for the sea surface but also for all the layers
underneath the surface. The simplest case is that where for the total duration T of
precipitation 27 is constant for 0 S t ^ T, while after the rainfall 27 = 0 for / ^ T.
In this case the solution for the total precipitation time T is

' = '*- (vSx)) ^'

and when precipitation has ceased

The maximum salinity disturbance q will reach by the end of the precipitation a value

^ ~ VipTrA) •

The salinity disturbance at the sea surface during the precipitation will follow the
equation

s* - s = q hr for (0 ^ r ^ T).

At the end of rain {t = T) q reaches a maximum value and then the disturbance
decreases according to the formula

_\l T \]\t

for (/ Z T).

Salinity of the Ocean, its Variation in Oceanic Space and in Time

161

The change in time of a salinity disturbance at the surface of the sea caused by precipi-
tation has the form shown in Fig. 67. Case 2 on Fig. 66 corresponds completely to the
theoretical solution as far as the observations allow a comparison. For the time /
required to reduce a salinity disturbance produced in a time T to a fraction q by
turbulence alone the above equation gives

for g = J
/ = 0-56 T

4 10'

3-52 T 24-95 T.

Thus a salinity disturbance produced by precipitation lasting one hour would fall
to one-tenth of its maximum value in about a day. Therefore, heavy rain can have an
appreciable effect on surface salinity and in a discussion of frequent rainfall this
circumstance deserves considerable attention.

Fig. 67. Change in time in salinity due to precipitation at the surface according to the theory.

In addition to the precipitation, the melting of icebergs which have drifted into
warm water can appreciably reduce the salinity in the remote and in the close surround-
ing waters. This process operates much more slowly than the precipitation but no data
for investigation are as yet available.

The physical process should not be so very different except that the limited extent
of an iceberg will confine it to a smaller space and it will thus have to be considered
in three dimensions.

2. The Horizontal Distribution of Surface Salinity

The most detailed charts for the Atlantic Ocean are those prepared by Bohnecke
(1936) based on all the available data. More recent charts for all the oceans have been
given by Schott (1928, and in improved form 1934); corresponding charts are also
given in his geography of the Indian and Pacific Oceans (1935). Plate 5 shows such a
chart on an equal area projection. The salinity of the open ocean varies between less
than 33%o in the north-eastern Pacific and a little more than 37%o in the horse latitudes
of the North Atlantic. The range of variations is little more than 5%o. All three oceans
have zones of maximum salinity in the subtropics with maxima of more than 37-25%o
in the North Atlantic and the South Atlantic. In the open northern Indian Ocean
the Arabian Gulf has maximum salinity values of more than 36-5%o in sharp contrast
with the low sahnity of the Bay of Bengal. In the southern Indian Ocean towards
Australia there is a subtropical oval region with a maximum salinity of more than
36-0%o.

162 Salinity of the Ocean, its Variation in Oceanic Space and in Time

In the Pacific the zonally oriented cores of maximum sahnity lie between 30° and
20° N. with somewhat more than 35-6%o and between 15° and 25° S. with about
35-6%o. Between the areas of the subtropical salinity maxima there is a belt of low
salinity for all three oceans located in correspondence with the region of the equatorial
counter currents.

On the polar side of the subtropical maxima in salinity there is a rapid decrease in
salinity which is particularly pronounced in the Southern Hemisphere in all three
oceans as far as the southern oceanic Polar Front (45°-50° S.). On the polar side of
this the salinity remains everywhere a little less than 34%^ especially in the area of the
Antarctic pack ice and drift ice. In the North Atlantic, due to the effect of the Gulf
Stream and the Atlantic Current, there is a sharp difference between the eastern and
western sides. In the eastern part there is only a slow decrease in salinity towards the
north; in the western part shows a belt of low salinity (less than 32%o) associated with
the Greenland and the Labrador Current which borders with a strong salinity gradient
the warm, more saline Atlantic water.

Table 73. Factors increasing or

decreasing the surface salinity.

Increasing

Decreasing

= evaporation

P = precipitation

= ice formation

/„ = ice melting

C+

= surface circulation (advection

C- = surface circulation (advection

of more saline water)

of less saline water)

M+

= mixing with more saline deep

M^ = mixing with less saline deep

water (turbulence and convec-

tion)

= Solution of salt deposits (Gulf

R = Inflow of fresh water from the

of Suez, Suez Canal, Gulf of

land (rivers, glaciers, icebergs)

Akaba)

(run off).

Table 73 shows the factors listed by Wust (1936), which increase or decrease the
salinity at the surface of the ocean. For a stationary distribution of salinity the effect
of all these factors at any point must balance. An analysis of the horizontal distribu-
tion of salinity in this way is not yet possible at the present time. However, if only the
mean meridional distribution in the space between 40° N. and 50° S. is considered
the above factors are considerably reduced so that a good correlation equation of the
form

S =f(E - P, C, M)

could be expected. At first attempts have been made to determine the dependence of
the salinity on the quantity (E — P) from the available dita. Recent calculations of
this type have been mad.' by WiJST (1930, 1936). Table 74 gives values of S, T and
CT< separately for the three oceans and for the total ocean. Values for E — P are given
later on in Table 87 (see Chap. VII, 3). Figu-e 68 shows the close relationship between
the distributions of S and {E — P). It has been found by accurate analysis that the
correlation equation 5" — f{E — P) for the entire ocean is linear:

70°N.-10°N.: S = 34-47 + 0-0150 {E - P) ± 0-1 l%o,

60° S.-10° N.: S = 34-92 + 0-0137 (E - P) ± 0-09%o.

Salinity of the Ocean, its Variation in Oceanic Space and in Time

163

36-5

•v

360

•-,

35-5

20
§

350

. 0

> /

34-5

lii

-20

340

-40

335

)

-60

330

-80

32 5

60°
N

40° 20°

20°

40° 60°
S

Fig. 68, Mean meridional distribution of evaporation-precipitation {E-P) and surface salinity
for the entire ocean (according to Wiisr, 1954).

Table 74. Mean values of salinity, temperature and density for 5° latitude zones for each

ocean and for the total ocean including adjacent seas

(WusT, 1954)

Zone

Atlantic Ocean

Indian Ocean

Pacific Ocean

Mean for all oceans

°lat.

5(%o)

r(X)

5(%o)

S(°D

r(°c)

<'t

5(%„)

N. 70-65

(33-5)

2-l)»

(26-79)t

(30-0)*

(-0-6)*

(24-12)

(33-4)

(2-1)*

(267 l)t

65-60

(32-45)»

(4-4)

(25-73)

—

(32-0)

(0-8)

(25-67)

(32-35)

(3-7)

(25-73)*

60-55

32-90

6-6

25-83

—

32-37

3-6

25-76t

32-66

5-2

25-84

55-50

34-56

8-8

26-82t

—

32-63

5-8

25-74

33-41

26-19t

50-45

34-80

11-4

26-53

—

32-98

7-7

25-74

33-69

9-2

26-08

45-40

34-90

14-9

25-94

—

33-53

11-8

25-51

34-14

13-2

25-70

40-35

36-47

19-3

26-08

—

33-98

16-2

24-93

35-11

17-6

25-48

35-30

36-9It

21-5

25-89

—

34-49

19-8

24-45

35-50

20-5

25-03

30-25

36-75

23-5

25-13

(39-57)t

25-6*

26-63t

34-95t

220

24-20

35-76t

22-7

24-62

25-20

36-74

24-8

24-74

36-92

26-2

24-44

34-90

24-4

23-47

35-C6

24-6

23-98

20-15

36-22

25-7

24-04

35-27

26-8

23-00

34-61

26-0

22-76

35-14

26-0

23-16

15-10

35-90

26-2

23-67

35-13

27-5

22-67

34-20

27-0

22-14

34-76

26-9

22-59

10-5

35-18

26-7t

22-98

35-12

27-6

22-63

34-04*

27-5t

21-85*

34-43*

27-4t

22-18*

5-0

3501 •

26-6

22-87*

35-07*

27-8t

22-49*

34-54

27-4

22-27

34-73

27-2

22-47

N. 70-Ot

3545

18-87

2511

3538

2718

22 90

3417

21-46

23 51

3471

2106

24-01

S. 0-5

35-65

25-5t

23-69*

3501

27-6t

22-55

34-91

27-Ot

22-67*

35-07

26-9t

22-85*

5-10

36-04

25-0

24-14

34-83

27-3

22-51*

35-20

26-6

23-01

35-25

26-5

23-09

10-15

36-65

23-9

24-94

34-62*

26-7

22-58

35-45

26-0

23-39

35-42

25-8

23-42

15-20

36-66t

22-7

25-26

34-93

25-3

23-22

35-65

24-9

23-88

35-62

24-6

23-96

20-25

36-34

21-7

25-34

35-34

23-4

24-09

35-70t

23-3

24-39

35-74t

23-0

24-51

25-30

35-98

20-6

25-37

35-69

21-2

24-98

35-53

21-2

24-86

35-68

21-1

24-99

30-35

35-53

18-4

25-59

35-81t

18-4

25-81

35-17

18-7

25-24

35-46

18-5

25-52

35-40

34-97

15-4

25-65

35-43

15-4

26-24

34-73

15-8

25-60

3504

15-6

25-89

40-45

34-42

11-0

26-34

34-66

11-2

26-49

34-51

12-8

26-07

34-54

11-8

26-29

45-50

34-07

6-6

26-76

34-07

6-3

26-80

34-24

9-6

26-48

34-14

7-7

26-58

50-55

33-87

3-0

27-01

33-85

2-9

27-00

34-12

6-6

26-80

33-96

4-4

26-94

55-60

(33-88)*

(0-5)*

(27-19)t

33-88

(0-7)*

27-18t

(34-02)*

(3-2)*

(27-1 l)t

(33-94)

(1-7)*

(27-18)t

S. 0-60t

35 31

16 07

25 63

34 84

16 08

25 08

35 03

19 64

24 65

35 03

17 99

24 99

* Minimum; t Maximum; J Without polar zones; () Approximate values.

164 Salinity of the Ocean, its Variation in Oceanic Space and in Time

For the individual oceans the deviations from a Hnear form are larger and Wiist
was able to show that these were due in the first place to mixing of the surface layers
with the layers underneath.

This linear dependence although unequivocal cannot be taken as a casual physical
relationship. This is readily seen, since if the surface salinity in an area was dependent
only on the difference evaporation-precipitation the constant excess of evaporation
(for always positive E — P) would cause it to rise continuously and a linear correla-
tion could not be maintained. The simple linear dependence is only a part of the gener-
ally applicable equation S = f{E — P, C, M) and to this equation adds the varying
effect of advection and mixing (Defant, 1931). This influence enters into the above
equation partly in the coefficient of the {E — P) term and partly in the first term which
represents primarily the effect of vertical mixing. If surface water of salinity S is
mixed with water of constant salinity 5*0 then the change of salinity due to mixing will
be proportional to Sq — S. The change of salinity due to processes of evaporation
and precipitation will be proportional to E — P. Under stationary conditions the
local change in surface salinity will be zero. Thus

^4 = <S-So)-i-b(E-P) or S = So + k{E-P).
ot

As shown above, this formula has been confirmed empirically and this mixing in
general proceeds with water masses of mean salinities of either 34'47%o or 34-92%o.
These values are mean values for the salinity at 400-800 m (subpolar intermediate
water). The fact that the value Sq is somewhat different for the individual oceans, as
Wiist has shown, proves the correctness of this assumption. The North Atlantic
north of 20° N. possesses a markedly high salinity which can be explained by the
absence of the weakly saline subantarctic intermediate water at a depth of 600-
800 m. The deep-reaching effect of the Gulf Stream and the strong inflow from the
Mediterranean exert by mixing a noticeable effect on the surface layer. Conditions in
the North Pacific are just the opposite. In contrast to the North Atlantic there is in
the North Pacific a well-developed subarctic intermediate current at 600-800 m,
which has its origin in the cold adjacent seas with a low salinity in the north-western
Pacific Ocean. The strong negative anomaly in the North Pacific is certainly associated
with this, because subtropical and adjacent seas are missing and therefore no inflow
of water with high sahnity can occur. The South Atlantic and the South Pacific with
no adjacent seas and well-developed subantarctic intermediate water show similar
but almost normal conditions. The difference £" — P is, however, always of decisive
importance, and since it is closely related to the general atmospheric circulation it
is clearly understood that the general outlines of the mean surface salinity must be
controlled by the atmospheric circulation.

Returning to the horizontal charts, an understanding of all the salinity details in
these charts involves not only the vertical mixing process with the layers underneath,
but also all the other factors influencing the surface salinity distribution. It is, however,
the oceanic and the atmospheric circulation that determine the details of the hori-
zontal distribution of salinity. The factors "solution of salt deposits" and "inflow of
fresh water" play no particularly far-reaching partf although the last factor (R) of
Table 73 has some importance in coastal regions.

Salinity of the Ocean, its Variation in Oceanic Space and in Time

165

3. The Vertical Distribution of Salinity (in Vertical Profiles and Sections)

(a) General Conditions

An increase of salinity with depth is not a necessary condition for vertical stabihty
in the ocean, since in general the temperature decreases so rapidly that static stabihty
is assured. In actual fact the highest values of the sahnity in the individual oceans are
found at the surface or in the uppermost layers and usually a decrease of salinity down-
wards. Figure 69 shows the vertical distribution of salinity down to 4000 m for the same
station as in Fig. 52. From 40° N. to 50° S., i.e., in the troposphere, S decreases rapidly
below a more or less homo-hahne surface layer of varying thickness. The strong

34935-0 2 4 6

S. %. SOO 2

4 6

8 34468 35-0 2468 36 0 2 | 4 6 8 9

340 2 4 6

340 2

4 6 8 1 ^^ ^ ? ^^° 2 4 6 8 ,360 2 \ b4-9350

: T I I .

>^ ' y ' ' )f ^

' \

400
800

""^

- ) V '' "^

\ I

\ /

1200
1600
2000
2400
2800
3200
3600
4000

\\^

,^^

)

]

L ;

I'M

1 1 1 1 1 1 1

2 4

7 8 .

50 i

i 1

5 1 1 1

Fig. 69. Vertical salinity curves for a series of oceanographic stations along a meridional
section through the Atlantic (corresponding vertical temperature curves are shown in

Fig. 52).

Footnote from opposite page

t This is demonstrated by the low salinity of the adjacent seas with a strong freshwater inflow
(such as the Black Sea and the Baltic a.o.) and can also be seen in coastal areas where there is a large
fresh-water inflow. The effect of these frequently turbid river v/aters is often found surprisingly far
out at sea. Charts of the mouths of the major rivers (Amazon, Congo, Tajo, La Plata) usually con-
tain a limited area in which the lighter water shows at the surface on top of the heavier sea-water;
but this is usually only the case in a thin layer and already in the wake of a ship the sea-water of much
more blue colour may be brought to the surface. An investigation of the mixing of the lighter river
water and the heavier sea-water at the mouth of a large river would be of some interest. The Suez
Canal shows the great effect on the salinity of solution of a salt deposit, in this case at the bottom of
the great Bitter Lake which is connected by the canal with the Mediterranean and with the Gulf of
Suez. Water of lower salinity flows in from both sides and causes a progressive dissolution of the salt
deposit and maintains in that way the high salinity of the water above at 50%o at the surface and
56%o at the bottom (about 10 m depth). Since the canal was first built (1869) when the water depth
was 7-56 m dissolution of the salty canal bottom has increased the depth here linearly to give a depth
in 1921 of 11-7 m. At the same time the salinity of 68%o in 1872 had fallen to about 52%o by 1924.
The available and, in parts, sparse data on the distribution of salinity in the Suez Canal and on the
currents caused by it have been dealt with by WiJST (1934, 1935) in two interesting papers.

1 66 Salinity of the Ocean, its Variation in Oceanic Space and in Time

decrease in temperature in these layers is thus associated with a strong decrease in the
salinity, This extends down to about 800 m where the salinity reaches a minimum of
34-3-34-9%o. There is then a second increase to about 34-8-34-9%o at about 1600-
2000 m and then a further slow decrease is generally observed down to the bottom.
The inversion in salinity at 800-1000 m becomes weaker and weaker towards higher
northern and southern latitudes, and from the polar fronts of both hemispheres
towards ^he poles it is entirely missing; the vertical differences are then small with
usually a slight increase in salinity if fresh water has not been added to the surface
layers by the melting of ice, but this becomes weaker and weaker towards the poles.
In contrast to this vertical distribution generally found, the North Atlantic shows a
pronounced peculiarity in middle latitudes which can be seen at some of the stations
in Fig. 69. The intermediate salinity minimum at about 800 m is missing here, and
from the core of upper layer of high salinity situated in middle latitudes the salinity
decreases almost uniformly down to the bottom. There is thus a marked asymmetry
between North and South Atlantic vertical distributions of salinity.

ib) The Salinity of the Oceanic Troposphere

The vertical distribution of salinity in the troposphere layers of the subtropics and
the tropics is worth a somewhat more detailed description. It has, of course, been
investigated more closely in the Atlantic (Defant, 1936). Almost all stations in the
tropics and subtropics show a nearly homo-haline top layer. Its thickness is not the
same as that of the thermal top layer but is usually somewhat smaller. In many cases
just below the quasi-isothermal top layer, however, still in the upper part of the ther-
mocline, there is a more or less well-developed salinity maximum. This maximum is
one of the most characteristic phenomena of the vertical salinity distribution of the
upper troposphere. Figure 70 shows an example of this. The "Meteor" 256 station
shows the maximum particularly well developed ; in a thin layer from about 50 m the
salinity rises from about 36-1 to 37-0%o and then falls again to the previous value. It is
worth noting that the salinity maximum appears there where the first drop in tempera-
ture occurs beneath the isothermal surface layer and not at about the maximum
temperature gradient of the thermocline (see Fig. 71). The sahnity maximum thus
extends just above the thermocline, but does not fully coincide with the density transi-
tion layer, the position of which is in turn fixed by the high salinity value. Careful
investigation of this sahnity maximum in the tropical and subtropical regions of the
Atlantic has shown that it is almost always present. Starting from the extensive sub-
tropical accumulation of very saline water (at about 25° S. and at about 30° N.),
where in a top layer down to the thermocline a homo-haline structure is found, a thin
layer of maximum salinity spreads out northward in the Southern Hemisphere and
southwards in the Northern immediately above or directly inside the thermocline.
This spreading occurs below the upper part of the top layer, in which salinity decreases
in both hemispheres towards the equator.

From this it can be concluded that the layer of the salinity maximum is formed from
the lowermost parts of the subtropical high salinity water by currents flowing towards
the equator. It thus represents the intrusion of highly saline water under the surface
layers of lower salinity of the equatorial regions and forms a part of the upper tropo-
spheric circulation.

Salinity of the Ocean, its Variation in Oceanic Space and in Time 167

200

300

400

500

34-5 350

355 360

i" /OO

365

370

Fig 70 Vertical temperature, salinity and density curves for the troposphere at "Meteor"
Stn. 256 (0 = 2-4° S., A = 39-3° W.).

Increasing values of 5 and /

Fig. 71, Position of the tropospheric salinity maximum relative to that of the thermocline

(schematic).

168

Salinity of the Ocean, its Variation in Oceanic Space and in Time

While the entire area between the behs of subtropical highly saline water in the
Northern and the Southern Hemisphere show these salinity maxima, just above or
inside the thermocline two belts without maxima stand out sharply; one between 10°
and 15° N, and extending from 45° W. eastwards to the African continent and a
second, but more narrow belt, between 2° S. and 3° S. and extending from 30° to 10°,
which is particularly well developed in the central part of the Atlantic Ocean (see
Fig. 72). These two belts without salinity maxima more or less mark the southern

Fig. 72. Distribution of salinity in the tropospheric salinity maximum in the subtropic and

tropics of the Atlantic.

and northern limit, respectively, of the subtropical water masses spreading towards
the equator. Between these two belts from about 7° N. to the equator the maximum
appears again and may be very pronounced. This is the region of the Equatorial
Countercurrent which is fed at a depth of 80-100 m from regions west of 35°-40° W.,
which are situated outside the area with no maximum. The salinity maxima of the
tropics and the subtropics are thus very closely connected with the tropospheric
circulation in these areas. The best illustration of the formation, extent and intensity
of this very pronounced thin layer of high salinity lying between low salinity layers
(above and below) is given by a vertical cross-section along the core of the Equatorial
Countercurrent and the Guinea Current in the Atlantic Ocean. This section is shown
in Fig. 73. It starts in the central Atlantic at about 18° N., 37° W., proceeds south-
wards to 10° N., 38° W. and then along the core of the Equatorial Countercurrent,
finally reaching the inner Gulf of Guinea. The layer of maximum salinity spreads
southward from the homo-haline top layer of the subtropical North Atlantic below
the low salinity surface layer towards equatorial latitudes. If the 35-5%o isohaline is

Salinity of the Ocean, its Variation in Oceanic Space and in Time

169

< C

•S 3
e «>

2 in

i o

h «

(53 C

^ J3

4) ^-

§2

a" „

c.S
o c

o p

1 70 Salinity of the Ocean, its Variation in Oceanic Space and in Time

taken as the upper and lower limit of the layer with maximum salinity it has an average
vertical thickness of only 50 m ; it stays about the same thickness over its long course
to well within the Gulf of Guinea, and the salinity of the core layer changes very little
after it has lost its tongue-like form along the first half of its route. A comparison of the
salinity section with a corresponding density section shows that the position of the
salinity maximum along the greater part of the cross-section coincides with the
strongest vertical concentration in the density field. The very saline water thus extends
in a thin layer along the thermocUne itself. The spreading in this layer is caused by
advection and turbulence but the latter factor must be of very little effectiveness, be-
cause of the almost unchanged character in this remarkably thin layer over such large
distances. It must be supposed that above and below the thermocline the transport of
water with maximum salinity is accompanied by strong mixing with the water above
and below, but that in the thermocline itself the stability strongly suppresses turbulence,
so that the almost horizontal spreading takes the character of a laminar flow. This has
been confirmed by calculations of the vertical exchange coefficient in the area of the
Equatorial Countercurrent by Montgomery (1939), who found /i^ = 0-4 g cm~^ sec~^
along the axis of the Countercurrent. Since lateral mixing was neglected in these
calculations the value found will be a maximum value; the true one must approach
rather closely the molecular diff'usion coefficient for salt in water (0-011). As men-
tioned above, the spreading must therefore be of laminar character. However, in
horizontal direction lateral mixing is very eff'ective and the lateral exchange coeffi-
cient Ay reaches the value of 4 x 10^ g cm~^ sec"S generally found.

From the deep-reaching accumulations of warm and saline water in the subtropics
there is not only a flow of this water towards the equator but also towards the poles
in somewhat deeper layers. Thus at depths only a little below the upper layer, and the
almost homo-haline top layer which shows decreasing salinity towards the pole, there
is a secondary maximum in the vertical distribution of the salinity. In the Southern
Hemisphere this poleward flow of highly saline water occurs first at a depth of 100 m,
but descending to a depth of 150 m or more, and continues on over a very broad
front across the entire ocean; however, the energy of this outflow is soon dissipated
and the maxima disappear due to mixing. In the Northern Hemisphere this maximum
is associated with the Gulf Stream and its continuation (the Atlantic Current) and it
can be followed across the entire Atlantic Ocean into the Norwegian Sea and further
polewards. Figure 74 shows a longitudinal salinity section given by Schott (1942)
through the Atlantic Current from the Wyville-Thomson Ridge past the Shetland
Islands as far as Spitzbergen. The Atlantic water soon descends underneath the cold
and low saline polar water of the surface layer. Although the salinity maximum is
decreased by mixing it can still be traced in the North Polar Basin and into the Barents
Sea. Its occurrence here was discussed in connection with the description of the vertical
temperature distribution in the North Polar Basin (see p. 133). An interesting and,
from the point of view of the method used, important study of this spreading of At-
lantic water {§ = 10-2°, S = 35-45%o) in the northern part of the North Sea, in the
Norwegian Sea and in the Barents Sea and its mixing with the surrounding water
{d = 2-5°, S = 34'90%o) made by Jacobsen (1943) should specially be mentioned
here.

From our knowledge of the tropospheric salinity maxima of the Pacific and the

Salinity of the Ocean, its Variation in Oceanic Space and in Time

171

172 Salinity of the Ocean, its Variation in Oceanic Space and in Time

Indian Oceans, we know their formation and spreading are still very pure. The much
stronger intensity of this phenomenon in the Pacific Ocean has been shown by several
recent oceanographic stations but detailed information about their extent is still
lacking.

The vertical salinity distribution in the stratosphere of the three oceans can best be
discussed by means of longitudinal sections through the Atlantic, the Indian Ocean
and the Pacific. The longitudinal section through the Atlantic Ocean is that given by
WusT (1936) through the Western Trough from the Weddell Sea to Davis Strait (see
Fig, 62). In the Indian Ocean a central section (Fig. 75) from the Antarctic to the south-
ern tip of India has been selected (Moller, 1929); the Pacific Ocean is characterized
by a vertical section through its eastern half (Fig. 76). In the northern part this section

34-0

35-5

35-0.

35-5 35-0.

BOCO

^000

■""^60* S 50° 40° 30° 20° 10° 0° N 10°

Fig. 75. Longitudinal salinity section through the central part of the Indian Ocean.

60° N

1000

2000

£ 3000

4000

5000

Fig. 76. Longitudinal salinity section through the central part of the Pacific Ocean,

Salinity of the Ocean, its Variation in Oceanic Space and in Time 173

is based on the "Carnegie" observations (Sverdrup) and south of 40° S. on the
"Discovery" observations (Deacon, 1937). Longitudinal sections through the western
and central parts of the Pacific Ocean have been given also by Wust (1929).

(d) Subpolar Intermediate Water

At 800-1000 m there is a characteristic lovv^ salinity zone extending across almost
the entire ocean though not always equally well developed. In the south it begins
always just south of the oceanic polar front where this special water mass sinks rapidly
from the surface to a depth of 800 m and spreads out from here with decreasing vertical
thickness and decreasing salinity in its core into the Atlantic across the equator to
about 20° N. It can still be traced north of here until it joins the deep and saline
water accumulations of the subtropics. There is little to be seen from an Arctic counter-
part to this subantarctic intermediate water. Only in the western section weak indica-
tions of such arctic intermediate water may be found as far as the Newfoundland rise.

Also in the Indian Ocean this intermediate water is found everywhere underneath
the high saline water mass south of the subtropics as an intrusion of low saline water
with its core somewhat deeper than in the Atlantic (approx. 1000-1200 m). In the
Pacific tongues of low saline polar water spread out below the high saline tropo-
sphere almost to the equator, from both north and south. The Antarctic branch of
low saline water forms just south of the oceanic polar front at 50°-60° S.; the arctic
branch formed in the area of the Okhotsk Sea is weaker; in the western and central
parts of the Pacific Ocean it can be followed to about 10° N. It is completely absent
in the whole of the eastern part of the Pacific and there is thus an asymmetry in the
salinity distribution similar to that in the Atlantic Ocean.

The vertical thickness of the subantarctic intermediate water is about the same in
all the three oceans (about 600 m) and it is separated from the troposphere above by a
sharp salinity (and density) transition layer. It is of particular interest that the inter-
mediate water is found with the same characteristics and thickness across the entire
transverse section of the ocean, especially in the Atlantic. Evidence for this is given in
Fig. 77 which gives a cross-section of salinity through the Atlantic at about 22° S.
This uniformity of this water across the total cross-section can be regarded as a conse-
quence of strong lateral mixing which leads to an equalization of all existing major
horizontal salinity differences.

A detailed investigation of conditions in the subantarctic intermediate water and
its meridional spreading in the Atlantic has been given by Defant (1936). The vertical
salinity distribution in successive cross-sections normal to the main direction of
spreading is best characterized by the dimensionless quantity {sq — s)I{sq — s^,
where ^o (=34-85%o) is the salinity which the subantarctic intermediate water takes on
by continuous mixing with the surrounding water and s,n (=34-19%o) is the salinity of
the subantarctic intermediate water in its region of origin before spreading out
towards the north. The quantity {sq — s,n) corresponds to a potential difference present
between the two oppositely moving types of water which is finally eliminated by mix-
ing. Determination of this quantity in cross-sections, 500 km apart from each other,
for the core layer (salinity minimum) and for several layers above and below this core
allows of construction of lines of equal values of the quantity (5'o — 5)/(^o~ ■5' m) expressed
in percentage of intermediate water. These lines then illustrate the mixing process

174 Salinity of the Ocean, its Variation in Oceanic Space and in Time

_ _ 20° i(r 0° 10'

Fig. 77. Cross-section of salinity through the Atlantic Ocean at about 23° S (profile VII at
24°-21-25° S., of the "Meteor" Expedition).

400
300

200
100
0
-100
-200
-300
-400
-500

95^-^89 83 ^79 76 737070
100 97 94 ) 87 83 80 79

82/72

95 /88 82 /7§ 74^7Cr^^67 68 68 65 /58 57 51 -50 47

^67 66 §u6C>57---..^6l &\^0 53^49 43 45 4L

^59 57 53-50'46'^>^53 50^— '49 43^.-^S8 34 36 ■ 33

u4034-

43 ^17

30—

5000

6000

7000

8000

9000

10000

000

12 000

Fig. 78. Percentage of subantarctic intermediate water in the core layer of this water type
the western side of the Atlantic. (Distribution of the quantity (^o — ^)/(*o — -yJ in per cent.)

Salinity of the Ocean, its Variation in Oceanic Space and in Time

175

along the entire spreading area for the entire western section through the Atlantic
Ocean (Fig. 78). This distribution has a clear similarity to that presented in Fig. 48
which shows the radial and turbulent spread of a particular water mass into surround-
ing waters. This distribution also corresponds to the processes of spreading in a so-
called "jet" (Freistrahl) (Prandtl, 1926; Tolmein, 1926; Ruden, 1933). Figure 79

100 X

*600 +800

1 ■

k u

■^

\
\

Fig. 79. Distribution of salinity relative to the minimum in the core layer of the subantarctic
intermediate water along the western side of the Atlantic for different vertical cross sections.

shows for each cross-section the distribution of salinity relative to the minimum in
the core and shows that the general distribution is the same for all cross-sections and
that the processes involved mu: t be essentially the same, geometrically and mechani-
cally, as in a "jet".

An accurate knowledge of the salinity values throughout the entire region of the
subantarctic intermediate water allows the vertical distribution of the quantity Ajpu
to be calculated from the equation on p. 106 for all cross-sections.

A rough calculation shows at once that vertical mixing and advection are able to
maintain the tongue-formed salinity distribution stationary in the subantarctic inter-
mediate current. The vertical salinity distribution at a distance of 8000 km from the
zeio point of the western section (about 13° S.) is as follows

-300

+ 200 +100

100

-200

-300 -400 (m)

salinity in %o

vertical gradient
(per 100 m)

34-71

0-54

0-42

O^S"

0-44

0-52

0-60

0-69

-017

-012

-004

+006 +008 +008 +009

Considering a vertical water column of 1 cm^ base between +250 and —250 m the
inflow of salt into the column from above and below is shown in Fig. 80, taking A =
4gcm-isec-^ The salt gain in the entire volume (5 x 10* cm^) thus amounts to
1-00 X 10"^ g/sec or 8-64 mg/day. Without an advective outflux this continuous gain
of salt would soon eliminate the salinity minimum of the subantarctic intermediate
water. Through the left-hand (southern) boundary of the water column (with an area
of 5 X 10* cm2) there enters an amount of salt of 5 x 10* X m • .s X 10"^ g, where u

176

Salinity of the Ocean, its Variation in Oceanic Space and in Time

is the velocity of the horizontal advection. At the right-hand (northern) boundary
there is at the same time an outflow of 5 X 10* X m-^-i x 10"-^ g of salt from the entire
volume so that the loss of salt in g/sec will be 50u{si — s). For a stationary salinity
distribution this loss must be compensated by a gain due to mixing, that is by
1-0 X 10"' g/sec. Taking u equal to 5 cm/sec, this is only possible for a horizontal
salinity gradient of(si — s) = 4 x 10^^° %o/cm in the current. The salinity at 7000 km

450 m

= 0-12 MO"

(vertical gradient
of salt at 450m)

950 m

I cm

068 X 10 g

Inc rease

salt content

1-00 X lO'^g

pro sec

salinity S
velocity U

salt flux

Fus« 10"

Fus, « 10

— ^ = 0-08 1 10
Az

(vertical gradient
of salt at 950m)

0-32 « 10 g

salt loss
Fu X I0''(s,-s)

Fig. 80. Salinity exchange and advection. For w = 5 cm/sec results salinity gain = salinity
loss: 5 X 10* X 5 X 10-^ (s^ - s) = 2-41 x 10"^ or {s^ - s) = 0-97 x 10-» "/oo per centi-
metre.

distance in the core of subantarctic intermediate water is 34'34%o, at 9000 km, however,
34-42%o, so that according to the observed values there is a salinity gradient of 0-08%o
for the 2000 km = 2 x 10^ cm. This gives exactly the value derived above of
4 X 10"^"%o/cm. The vertical and horizontal salinity distribution in the subantarctic
intermediate current at this point can thus remain stationary with values of
4 g cm~^ sec~^ for A and 5 cm/sec for u. The ratio

Aj pu = t — 0'8 cm/sec

satisfies therefore the condition of a stationary state of the phenomenon in time. It
is fairly easy to see that the above calculation gives only the quantity Ajpu and not the
absolute value of the individual quantities.

Salinity of the Ocean, its Variation in Oceanic Space and in Time

111

Using the above mentioned relationship the quantity Ajpu can be determined
numerically for every point more accurately than in the rough calculation made here
by deliberately selecting a large water column.

Table 75. Mean values for Ajpu for transverse sections {normal to the direction of
spreading) through the subantarctic intermediate water in the Atlantic

Position of

Vertical distance from the core (m)

Mean depth

sections

+300

+200

+ 100

-100

-200

-300

(m)

37°-30°S.

27°-21°S.
25°-12° S.
3°S.-2°N.

4°-8°N.

3-5
2-8
2-4
(2-2)
(2-3)

20
1-7
1-4
1-4
1-7

1-3
10
0-8
11

1-7

10
0-7
0-4
0-9
10

1-2
1-6
1-3
1-2
1-6

1-7
30
3-2
2-2
2-4

2-4
3 0
3-9
3-2
3-4

850
800

725
700
625

Mean

2-64

1-64

112

0-82

1-38

2-50

318

740

Table 75 shows that the vertical distribution of A I pu scarcely changes along the
entire region of spreading: Al pu is least at the core of the spreading water (on the
average 0-82) and rises steadily both above and below with the distance from the core
to large values (about 3). If the distance from the core axis is denoted by z, then
Alpu=^f(z).

When / is the Prandtl mixing length

and therefore

Az) =

Pcu

u dz

Integrating this equation for constant / from the core (2
below the core then, since dujdz is always negative,

f* u

/(zyz=-/Mn-,

0), to a distance b above and

where Mq and u are the values of u in the core and at a distance b from the core. The
value of the left-hand side can be found from Table 75 and this gives, knowing /,
the ratio of m/mq as a function of the vertical distribution and ihus also of the quantity
AJAq, where A^ and A are the exchange coefficients in the core itself and at a distance
b from it. The constancy of / can be tested on the plausible assumption that the velocity
of the current within a distance of ±300 m from the core falls to a low value; over a
wide region / is almost constant with a value of about 150 cm. The results of the
calculation are shown in Fig. 81. The relative distribution of velocity over a trans-
verse section has a striking similarity to that distribution found experimentally in
turbulent spreading processes in a "jet". This justifies the conclusion that the spread
of the subantarctic intermediate water in the Atlantic northwards, along the boundary
between the oceanic stratosphere and the troposphere, is very probably a process that

178

Salinity of the Ocean, its Variation in Oceanic Space and in Time

is largely equivalent to the phenomena in a "jet" CFreistrahl) at some distance from
the nozzle (Diise). The distribution of^ AjA^ in transverse section is also quite charac-
teristic. The maximum appears in the lower part of the spreading layer (150 m be-
neath the core) ; below this the ratio falls rapidly, but above only slowly. This striking
distribution of the exchange coefficients can be readily explained by the different
stability conditions above and below the core.

+WOm
1.2

o,e

0,0

+200

-200

-^00

1
A ^,

— ,i

// .

*s^^

Fig. 81. Relative distribution of the exchange coefficients, AlA^ and the current velocity,
w/«o along a cross-section through the subantarctic intermediate water along the western

side of the Atlantic.

(a) Salinity of the deep water below 1500 m. In the deep layers of the Atlantic the
salinity increases slowly from the Antarctic regions across the equator as far as the
deep-reach 'ng, warm and high saline water of the northern subtropics (20°-40° N.);
from here towards the north it decreases slowly in the upper layers. However, in the
deeper layers the increase to about 20°-40° N. is much less. The asymmetry of the
salinity distribution shown so strongly in the subantarctic intermediate water is also
present in the deeper layers but not to the same extent. This contrast is due in the
first place to the strong accumulations of saline water in the subtropics, but in these
layers it is also reinforced by the inflow of highly saline water from the Mediter-
ranean through the Straits of Gibraltar. Everywhere in this area there exists a well-
defined maximum in the vertical sahnity distribution at 1300 m (at about 20° N.)
lowering to 2500 m (at 35° S.) that must be attributed to the spreading of the
Mediterranean water. This effect of inflow from the European Mediterranean can
be seen particularly on the salinity chart for 1000 m depth. The spread of this type
of water will be discussed in greater detail later on (Vol. I, part 2, Chap. XVI, 3).

The nature of the water beneath the upper part of the stratosphere in the Atlantic
indicates an area of formation in higher northern latitudes (north of 50° N.) in the
Western Trough. Here it is formed at the surface during the late autumn and early
winter, sinks by thermo-haline convection to great depths and spreads out more or
less horizontally below 2000-2500 m to fill the lower part of the stratosphere. The
high oxygen content which characterizes this water type will be discussed later in
connection with the oceanic circulation (see Vol. I, part 2, Chap. XX, 7.

A similar contrast between the higher latitudes of both hemispheres is also present
in oceanic stratosphere of the Indian Ocean. Here it is due in the first place to the inflow
of highly saline water from the Red Sea. Coming from the Straits of Bab-el-Mandeb
(seep, 182 and Fig. 84), it sinks to about 10(X)m, mixes with less saline water in the Gulf of

Salinity of the Ocean, its Variation in Oceanic Space and in Time 179

Aden and from here extends southwards beneath the Antarctic intermediate water
at a depth of 1500-2000 m as a tongue of highly saline water. This salinity maxi-
mum shows very clearly throughout the western and central parts of the Indian
Ocean.

In the Pacific the few observations that have been made below 1 500 m show a
remarkably uniform vertical and horizontal salinity distribution at all latitudes. Its
average value is about 34-65-34-68%o, but it is nowhere connected with the equally
high values in salinity of the surface layers. There is no tropical or subtropical ad-
jacent sea acting as a source for saline water for the Pacific stratosphere like the
Mediterranean does for the Atlantic one or the Red Sea for the Indian Ocean strato-
sphere. It must therefore be supposed as pointed out by Sverdrup (1931), that the
Pacific deep water below about 1 500 m depth for which there is no area of formation
in the Pacific itself must be formed in the Indian Ocean or even in the Atlantic.
Water masses from these two oceans must be carried to the east by the Antarctic
ciicumpolar ocean current and then spread northward in form of current branches
to fill the deep basins of the Pacific.

(8) The salinity of the bottom layers. The salinity of the deepest layers shows also the
same characteristic distribution already known from the bottom temperatures. In
the Atlantic Ocean (Wust, 1936) it varies between 34-62 and 34-92%o in the most
northern parts; this is explicable from conditions of formation of the bottom water.
The deepest parts of the Antarctic regions are filled with Antarctic bottom water with
a salinity of 34-67-34-69%o, formed at the continental slope of the Weddell Sea
(see p. 14?). Above this the Antarctic deep water is found at 5000-4000 m with 34-62-
34-66%o that feeds the Antarctic bottom currents of the Eastern and Western Troughs.
The isohalines of meridional sections demonstrate a clear conformity with the
bottom profile and show the penetration of the water across the Equator in the Western
Trough and the Eastern Trough as far as the Whalefish ridge. Figure 82 gives meri-
dional salinity sections through the Western and Eastern Troughs of the Atlantic which
show how the spreading of the bottom water is reflected in the distribution of the
salinity in the same way as in the distribution of potential temperature (see p. 152)
deduced previously.

A typical Arctic bottom water cannot be recognized from the salinity distribution
though traces of it can be detected in the Labrador Basin north of the Newfoundland
Rise (WiJST, 1943). Our knowledge of the salinity of the bottom water of the other two
oceans is still pure due to a lack of systematic salinity data.

4. The Horizontal Distribution of Salinity at Particular Depths

Horizontal charts of salinity distribution are so far available only for the Atlantic:
they are given for instance in the ''Meteor'" Report for depths of 200-800 m at 200m
intervals, for depths of 1000-2000 m at 250 m intervals and for depths of 2000-4000m
at 500 m intervals. Plate 6 shows charts for 400 m and 1000 m depths. It is clear that
these charts do not give other information than the longitudinal and transverse sec-
tions. The charts down to 800 m, of which the 400 m chart is given as an example,
all show essentially the surface salinity distribution; only the horizontal differences
become smaller with increasing depth. Of the two extensive regions with salinity
maxima in the subtropics the northern is the larger. The highest values appear,

180

Salinity of the Ocean, its Variation in Oceanic Space and in Time

'mdaQ

Salinity of the Ocean, its Variation in Oceanic Space and in Time 181

however, not in the central part of the Sargasso sea but are displaced'in the peri-
pheral parts towards the west, partly on the right hand (north) side of the North
Equatorial Current (especially at 200 m) and partly on the right-hand side of the Gulf
Stream (especially at 400 m, but still visible at 1000 m). This distribution is a dynamic
effect of the currents which cause an enormous water transport.

Below 600 m the influence of the high salinity inflow from the European Medi-
terranean begins to appear and extends already at 800 m to 40° W. It remains the
principal phenomenon in all charts down to almost 2000 m and the remarkable
asymmetry between the North and the South Atlantic shows particularly clearly here.

Below 2500 m the horizontal salinity differences already become very small though
there is still a noticeable salinity gradient from north to south. South of 40° S. more
pronounced differences in salinity reappear which indicate the increasing influence
of the Antarctic deep and bottom water.

5. Salinity in Adjacent Seas and Sea Straits

In discussing the temperature distribution in adjacent seas (see p. 1 29) it was already
emphasized that beneath the sill depth in all the adjacent seas theie is an almost
constant salinity; in the adjacent seas without winter convection it is identical with
the salinity of the open ocean at the sill depth off the passage ; in the adjacent seas
with a winter convection, on the other hand, it is identical with the surface salinity
at the time of the thermo-haline mixing (see Tables 56-66).

When there are relatively large differences between the water masses of the free
ocean and those of the adjacent sea, the equilibration movements in the more or less
narrow sea straits connecting them show rather striking conditions which deserve
particular attention. The interchange of water between the European Mediterranean
and the Atlantic is a consequence of currents through the Straits of Gibraltar, which
carry water at the surface and in the uppermost layers into the Mediterranean to-
wards the east, but in the deeper layers beneath towards west. Corresponding con-
ditions are also found in the Straits of Bab-el-Mandeb, but in other sea straits the
thermo-haline structure imposes reversed flow conditions. In the Dardanelles and
the Bosporus, Aegean water flows into the Black Sea in the lower layers, while the
flow into Mediterranean occurs in the upper layers.

Similar conditions also prevail in the connecting straits between the North Sea and
the Baltic, where North Sea water enters through the Oresund and the Great and
Little Belts along the bottom, while contrary the surface water flows out of the Baltic.
All these water transports are associated with considerable changes in temperature and
salinity. It could hardly be expected that these processes should be stationary ones. In
fact they are turbulent and occur in pushes and therefore cause extremely large
variations in both factors that they can only be investigated and understood with the
aid of synoptic surveys. The available summarizing descriptions of the distribution of
the different oceanic factors in such straits should thus be interpreted with some
caution.

Figure 83 shows the distribution of temperature and sahnity according to Schott
(1928) through the Straits of Gibraltar for the transitional period from spring to
summer when average conditions prevail in the currents. The isohalines of the longi-
tudinal section show clearly that the highly saline Mediterranean water, for which.

182

Salinity of the Ocean, its Variation in Oceanic Space and in Time

8°7' 7°54' 7°20'

Albocoral66Mbwe2 Thor9l MSars28
•aS-Z? 211 BllO 210

7°0'W 7°30'

M.S(ys22Thor92aim.LoboXX[
210 SHO 223

6°0' S'SO'

Thor97 Almi.obaXXX2IIThor98fllmi.obi
3fflO 2123 --511021123

1500

Fig. 83. Temperature and salinity distribution through the Straits of Gibraltar at the transi-
tion from spring to summer (mean conditions, according to Schott).

due to mixing, a slowly westwards decreasing salinity with the surrounding water is
characteristic, sinks beneath the weakly saline Atlantic water below about 300 to
400 m. The temperature distribution shows identical conditions. This water continues
to sink to about 1000-1200 m off the Spanish Bay, and from here it spreads out into
the Atlantic as a more or less horizontal layer of highly saline water. The di'.tribution
within the strait shows strong seasonal variation : at the end of the winter the contrasts
are reinforced, at the end of the summer they are weakened, but there is always a
continuous outflow of water with a high salinity from the Mediterranean into the
Atlantic and the submarine ridge never forms a barrier to the Mediterranean water
as BuEN attempted to show (1927).

Conditions in the Straits of Bab-el-Mandeb are rather similar ("Schott, 1929).
The highly saline deep water of the Red Sea (S 37%o) flows over the sill at 150 m
depth north of the strait of Perim into the Gulf of Aden (Fig. 84). It sinks here to
500-1000 m and then spreads out horizontally at such a depth, in which the density
of the sinking water becomes equal to that of the surrounding water.

Also the transition from the higher salinity of the North Sea (about 32%o) to the
lower salinity of the Baltic (about 7%o) is not at all continuous, as one might easily
be misled by studying mean charts only, but usually occurs rapidly, mostly in
two steps (Wattenberg, 1941). The first rapid change occurs near the boundary be-
tween Skagerrak and Kattegatt and changes its position very little in time; the second
much sharper change has a more variable position between the southern edge of the
Kattegatt through the Great and Little Belts to the rises leading to the actual Baltic
(Darsser and Drogen Rises). These jumps in salinity have all the properties of true
hydrographic fronts. They separate three water types: North Sea, Kattegatt and Baltic
water. Figure 85 shows the distribution of the surface salinity from the Skagerrak to the
Baltic in three diff'erent cases, and illustrates clearly the typical distribution at these
fronts. The latter are not, however, stationary in location but move around continually

Salinity of the Ocean, its Variation in Oceanic Space and in Time

183

'mdaa

184

Salinity of the Ocean, its Variation in Oceanic Space and in Time

Kattegat

Baltic

Sk. L Jr.

AKn. L.Gr. Or.

Chr.O.

1
\

1 1 1
M.G.

\ \
\ \
\ > —

\
\

6.V.

\|6E \2.BZ:

1938

--^

— ^^''Nt-

\ \

\^ -V,

>62.

\l6.I2

N2.12
\
\

^\ \

1938

III 11

Sk. L.R. O.FI. Sch. Gr Ky. Hfi. K.N. RB. G.R.

Kattegat

Large ondF Belt

Baltfc

Fig. 85. Changes in surface salinity betv.'een the North Sea and the Baltic (from the Skagerrak

into the Baltic) in three cases (according to the individual values recorded on 2 and 16 April

and 16 May 1938, according to Wattenberg).

often at considerable speed in one or the opposite direction, and these displacements
are ttien associated with jump-like changes in T and S at any given point.

In the sea straits so far discussed the equalization currents are superimposed (one
above the other) and the water movements occur along a boundary surface sloping
in the direction of the strait. This superposition of the two types of water appears to
be causally associated with the narrow width of these straits. If this surpasses a cer-
tain value then the interchange of the different waters no longer takes place through
currents flowing one above the other, but rather side by side in the strait, whereby the
boundary surface now slopes transverse or normal to the main longitudinal axis of
the strait. This type of water interchange is apparently present in the straits between
the White Sea and the Barents Sea (Timonoff 1925), see Vol. I, Chap. XVI, p. 1-3
for a discussion of the dynamics of this process.

Chapter V

The Density of Water Masses in the

Ocean ^ Vertical and Horizontal Density

Distribution and its Stability

1. Diurnal and Annual Variations at the Surface

The diurnal and annual variations are uniquely determined by that of the tempera-
ture and salinity. Since the diurnal temperature variation is essentially parallel with
that of salinity, the effects of both factors on the density partly cancel each other out,
and apart from the fact that they are both small anyway, the diurnal surface-density
variation is thus a rather insignificant phenomenon. In general, the aperiodic changes
in density during the day are so large that they completely mask the regular diurnal
variation. At anchor stations the average diurnal variation in density, taken as the
average over several days, is of the order of 0-05-0-1 in a^ (Table 76),

Table 76. Diurnal density (of) variation at the ocean surface (Atlantic Ocean)

Anchor

Hours

Diurnal

stations

variation

"Meteor"

5° S.-5° N.

22 +

0-75

0-76t

0-76

0-74

0-71

0-67

0-65*

0-69

0-71

0-73

0-74

Oil

"Altair"

44-5° N..

34° W.

26 +

019

0-20

0-21t

018

017

016*

016

017

0 06

* Minimum; t Maximum

The maximum occurs in the morning or in the forenoon; the density then falls,
probably due to the rising temperature — and in spite of the increasing sahnity — to a
minimum in the afternoon ; the amplitude is everywhere very small.

The annual density variation is much larger and its amplitude usually is of the order
of 1 -00 and 2-00 in Of depending on whether the annual variation in the temperature is
parallel or inverse to the corresponding salinity variation. The annual density variation
can be conveniently presented by plotting the monthly values on a [rS'J-diagram. This
has the advantage of providing a visual impression of the variations in temperature and
salinity, and also in density. For annual variations in Tand S, following pure sine curves,
the annual variation in density will be shown on such a diagram as a straight line if the
annual variations of the two factors run either parallel or inverse. If the amphtudes are
normalized (choosing scales of equal length for T and S in the diagram) then the
straight line will be at an angle of 45° with the temperature axis, but for inverse

185

1 86 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution

variations of T and S (phase difference of 6 months) it will be at an angle of 135°.
For a phase difference of three months the density values will lie either clockwise
or anticlockwise around a circle. This method has been used by Neumann (1940)
for a close investigation of the annual density variation in the area of the Gulf Stream
north of the Azores. Figure 86 shows such annual density variations for some five-

Sfoo

340

332

•-^■•^/

///////

Ijjjjl

/ /vi / / / /

y/n

/ /Vs/ / /

-y/A

V////Z

V77?/T

West of New Fbundlond

50°-45''N

8 10

45°-50°W

35-6

2 13 T

12 14 16 .18

50°- 491^ 20°-25''W

Between New Fcxjndland and Azores

27-0 26-0

T' 14 16

45''-40°N

20 22 20

40°-45''W

14 16 18 20

45''-40°N 25° 30°W

Fig. 86. Annual density variation at the surface of the sea in the area of the Gulf Stream
north-west of the Azores (according to Neumann).

degree squares according to the above method. The amplitude is largest {Aa^ = 2-09)
at the boundaries of the Gulf Stream and the Labrador Current, then decreases to the
east and south-east to only 1-5-1 in o-^. The maximum occurs in late winter (February-
March) and the minimum without exception in August. In the western squares the
densities lie almost on a straight line inclined at an angle of 135° to the temperature
axis. The more or less sinusoidal annual variations in T and S show therefore a phase
difference of about six months.

Similar investigations for other oceanic regions are entirely missing. Bohnecke
(1936) has given a chart showing the annual variations in surface density over the
entire Atlantic. As may be seen from this chart in the large areas of the North and the
South Equatorial Currents the annual variation in ct< is generally less than I-O. It
rises locally above 1-5 only at the boundary between the North Equatorial Current
and the Equatorial Counter Current (about 10° N.). In the tropics and the subtropics
the annual variation is on the whole large only in those areas, where there exists a
large annual variation in salinity (mouth of the Amazon, Gulf of Guinea, region with
upwelling water east of Cape Verde Island). In higher latitudes the annual density
variation remains, in general, also between 1-0 and 1-5, only falling below 1-0 north of

Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 187

50° N.; however, in regions close to the coasts seasonal displacements of different
types of water also cause large annual density variations (>20).

2. Density Distribution at the Surface of the Ocean

It is very characteristic of the density distribution at the surface of the ocean that in
spite of the extended strong salinity maximum in middle latitudes there is a rather
regular increase of density from the equatorial regions towards the poles in all oceans.
This already points towards a decisive influence of the temperature. Figure 86 shows
the distribution of density at the surface of the Atlantic Ocean according to Bohnecke
(1936). This picture illustrates the meridional increase from about 23-0 at 7°-8°N.
to a value somewhat larger than 27-0 in higher latitudes mentioned above. Table 77
gives mean values for successive latitude zones of 5 degrees width. The increase is not
entirely uniform in all these zones ; the regions of subtropical convergence stand out
as zones with a smaller density gradient and this gradient becomes larger again only
near the oceanic polar fronts. Beyond the extensive areas of maximum density in
subpolar and polar regions of maximum density the surface density seems again some-
what to decrease.

Table 77. Mean meridional density distribution in the Atlantic (o-^)

Latitude

0°

10°

20"

30°

40°

50°

60°

70°

Northern Hemisphere
Southern Hemisphere

23-50
23-50*

23-28*
24-53

24-48
25-31

25-44
25-42

25-90
26-06

26-69
26-75

27-25t
27-15t

26-61
26-93

* Minimum; f Maximum

For the Indian and the Pacific Oceans the surface density charts of Schott (1935)
give only summer conditions for each hemisphere. These charts show essentially the
same basic features as in the Atlantic. In the northern Indian Ocean only, conditions
are somewhat complicated due to the large annual variations in salinity. The large
differences in density between the Bay of Bengal with values of 22-0-1 8-0 and the
Arabian Sea with an increase to 23-0 or even to 24-0 should particularly be mentioned.

3. Vertical Density Distribution and Horizontal Charts for Different Depths

The density is equally expressed by the quantity a, for the deeper layers. In this
quantity the effect of pressure acting on the water mass is not taken into consideration
and it refers therefore to zero sea pressure. As a rough approximation, ct< can be taken
as the density which would occur in a water mass after displacement ofthe mass with its
in situ temperature and salinity from the depth to the surface (potential density);
thereby only the adiabatic temperature effect remains out of consideration.

For a study ofthe vertical density stratification ofthe ocean it is necessary to go back
to the values of the density or the specific volume in situ. Table 78 contains values for
a standard sea at 0°C and 35%o salinity, the vertical distribution ofthe density 0-^,^,5,
and of the specific volume a^.^^p, and the corrections which must be applied to these
as,<,^ to obtain the distribution at 35%o for 10° and 20°C, respectively, or at 0°C for
32-5 and 37'5%o, respectively.

188 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution
Table 78. Density and specific volume for dijferent s, t, p (ct^, j, j, and a^, <, p)

Depth

(m)

Pressure

Density
o„ 1, ,
(OX,
35°/oo)

Specific
vol.

"it 1. V

(0 C,

35»/„o)

357.0

(

0°C

dbar

10°C

20°C

32-5»/oo

37-5»/oe

2813

28-23
28-36
28-61

29-08
29-56
3003
30-50

32-85
37-52
41-09
46-40
50-72

0-97264

253
242
219

174
129
084
040

0-96819
388

0-95970
566
173

+ 109

+ 0 X 10-'

+ 318

+ 0 X 10-'

1 -191 - 0 X 10-'

+ 190 + 0 X 10-*

100

200
300
400
500

1000
2000
3000
4000
5000

+ 0

+ 1
+ 2

+ 4
+ 7
+ 9
+ 11

+ 21
+ 41
+ 60

+ 77

+ 1

+ 2
+ 3

+ 7
+ 11
+ 14
+ 17

+ 34
+ 66

- 0

- 1

- 2

- 4

- 7
-11
-14

-17

+ 0
+ 0
+ 0

+ 1
+ 1
+ 1
+ 2

+ 4
+ 7
+ 11
+ 14

+ 17

This type of presentation was chosen in order to allow differences from the values
for standard ocean to stand out. The correction terms enclosed by rectangles refer
to the quantities already considered during the determination of o-^ and a^. It is
obvious that these are the main correction terms. It is, however, generally customary
to judge the vertical density stratification from the a^-values. This will also be done
here and the more correct cr^^f^j, and as,t,p will be considered again later.

From stability considerations it is to be expected that the values of cr^ will increase
with depth. Apart from the surface layer down to about 50-100 m, this is always the
case. In the tropics and subtropics the increase is characterized by a transition layer
which begins just beneath the top layer, rising to a maximum gradient, then slowly
changing towards the deeper layers to a much smaller gradient. Towards higher
latitudes the intensity of the transition layer decreases more and more and beyond
35° N. and S. it becomes of no significance. In the Atlantic, for example, it can then
scarcely be regarded as a transition layer. In these regions the vertical density gradient
decreases steadily from the surface value downwards. In polar and subpolar regions
the density gradient from the surface layer down to the sea bottom becomes minimal.
Figure 87 shows the vertical distribution of o-j for some stations for which the T-
and ^-distributions were given already in Figs. 52 and 69.

In the uppermost layer a small increase in the a^-values with depth can occasionally
be noted (see the first three stations of Fig. 87). This does not, however, necessarily
mean that the vertical density stratification of these water masses is unstable. Because
reduction of the a^-values to the more correct Og^f^p may remove these small differences
as happens in the three cases in Fig. 87. There still remains, however, a large number
of stations where there is undoubtedly a state of weak instability (see Chap. V, 6).

A better insight into the nature of the vertical cr^-distribution through the entire
ocean is given by constructing longitudinal sections. Wiist has prepared sections of
this type for the Atlantic, indeed he chooses the same sections as for T and S (see
Fig. 62 p. 146 and 147). Figure 88 presents the o-rsection along the Western Trough of
the Atlantic ; the others show in principle the same picture. Although at the surface there
is a general slow increase of density, from the equatorial zone towards high southern
and northern latitudes, already at 100 m depth and below a diflferent distribution is

Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 189

T 25 26 27 28 of

23 24 25 26 27 28 27 28

500

1000

1500

2000

2500

3000 -

3500

4000

/ . ' 1 ,- \

i. '

\:~-.

\ 1
\
\

\ \
\ \
\ \

\ \

2j 3|

1 1 1

26 27 28

Fig. 87. Vertical distribution of the density cr^ at some oceanographic stations in the Atlantic:

"Meteor" 254

2° 27' S.

34° 57' W.

"Meteor" 170

22° 39' S.

27° 55' W.

"Meteor" 8

41° 39'S.

30° 06' W.

"Meteor"

Greenland 122

55° 03' N.

44° 46' W.

found that resembles more closely that of the temperature at these depths. In the
subtropics of each hemisphere the lighter water extends down to great depths while in
the equatorial zone the heavier waters of the deeper layers extend higher upwards
to just below the strongly developed density transition layer. This gives rise to a
horizontal density gradient from the equatorial zone towards the two subtropical
regions, that is opposite to the surface gradient. This gradient remains unchanged in
direction, though becoming weaker and weaker down to about 2000 m below which
the meridional density differences are usually rather small. In all the vertical sections
there is, however, a weak density gradient from high northern latitudes across the
equator to as far as 40-50° S. which is connected with the oceanic circulation of the
deeper layers.

It is readily understood that horizontal charts of a^-values show in principle the same
picture. A comparison of such charts with charts of the relative topography of the
isobaric surfaces (Helland-Hansen and Nansen, 1926) demonstrate that the course
of the isopycnals on the horizontal charts is in essential agreement with that of the
dynamic isobaths. The horizontal circulation of the water masses can thus be deduced
approximately from the horizontal distribution of the o-^-values. In that way stream
lines for the relative water flow are obtained (i.e., with reference to the lower layers).
Arrows showing the direction of flow are thus often inserted on the isohnes on
isopycnic charts of the upper layers to indicate the currents. These are subject to the

190 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution

000/1
00091

O <N

H <L>

C tn
O u

':3 '-'
-B

•2 S
^ &

>% "^

g-a
^ g

'5b

c
o

H-1

uj 'mdao

Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 1 9 1

0° 70° 60° 50° 40° 30° 2(f W° 0° m° 20° 30° 40° 50°

J20° ^110° ^ 100°^ 90[

Fig. 89. Density of sea water cr, at the surface of the Atlantic (according to Bohnecke).

1 92 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution

rule that in the Northern Hemisphere the higher density values are found to the left
of the direction of flow, while in the Southern Hemisphere they are found to the
right (see Fig. 89).

Horizontal charts of the density for 400 and 1 500 m in the Atlantic Ocean have
been given in Plate 7 to supplement the above brief remarks. The first chart shows the
Gulf Stream system very clearly by the strong concentration of the isopycnals into a
narrow belt running from the Gulf of Mexico through the Florida Straits to the New-
foundland Banks and beyond to the north-east. Compared with this very large hori-
zontal density gradient, that connected with the equatorial currents is only very
small. In the Gulf Stream region the 400 m chart indicates another phenomenon that
is characteristic of stronger gradient currents and is apparently missing in pictures
of the surface current. On the right-hand side of the Gulf Stream some isopycnals
deviate outward and turn into a south or south-west direction, opposite to the direction
of the narrow band surrounding a strong longish density maximum at the right-hand
side of the current core. These backward-turning isopycnals indicate the presence of a
countercurrent to the right (to the east) of the Gulf Stream which is of considerable
importance for the dynamics of this ocean current near the American coast.

In the Southern Hemisphere the isopycnals are strongly concentrated in the regions
of the Agulhas Current, the Brazil Current and the Falkland Current. In addition, a
steady rise of density exists in the Southern Hemisphere extending around the entire
southern ocean which is associated with the broad circumpolar West Wind Drift of
the higher southern latitudes. All density charts down to 800 m show very much the
same picture, though the density gradient becomes gradually smaller and the density
maxima of the subtropics are thereby somewhat displaced towards the poles. At first,
a different distribution begins to appear below 1000 m, which dominates in the
1 500 m chart. This is the density gradient from high northern latitudes to the mini-
mum zone between 35° and 40° S. This north to south density gradient becomes less
and less pronounced with increasing depth and below 4000 m the horizontal density
differences become already very small.

4. Potential Density and Isentropic Analysis

In earlier times potential density was considered a significant property on which to
form an opinion about the state of vertical equilibrium of oceanic stratification. As
already stated (see p. 1 88) potential density is calculated from the in situ salinity and
the potential temperature. Since the latter differs only at great depths from the in
situ temperature and then by only a few tenths of a degree centigrade, the difference
between o-^ and a^ remains very small and is almost insignificant as shown in Table 79.
It thus makes little difference whether the vertical density distribution is judged by
means of the customary a^ or of the more correct oq. The potential density has recently
become of greater interest due to the introduction of the method of isentropic analysis.
In meteorology, the investigation of the distribution of individual meteorological
elements on surfaces of equal entropy has been modernized and this has led to ap-
preciable success. Parr (1938, 1938^) has studied the spreading of oceanic water types
in a similar way by following the changes in salinity and temperature on surfaces of
equal density ct<.

Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 193

Table 79. Density a^ and potential density oq at ''Meteor'' station 310
(19-3° N., 25-0° W.)

Depth

Temp.

Salinity

f^e

CT0 — O,

("0

CC)

(%o)

(density)

(potential
density)

(difference)

21-45

36-69

25^68

25-68

000

21-35

-65

■68

-68

-00

20-27

•77

26-05

-00

20-01

■795

•14

-14

•00

100

19-53

■75

-24

-25

•01

200

15-83

-11

•65

-66

•01

300

13-63

35-74

-84

-85

•01

400

11-69

-44

2700

27-01

•01

500

10-58

•36

-16

-17

•01

600

903

-14

-24

-25

•01

800

7 00

34-94

-39

•40

•01

1000

604

-96

-53

-54

•01

1500

4-50

35-03

-77

-79

•02

2000

3-55

34-965

-82

•84

•02

3000

2-83

-93

-87

•89

•02

4000

2-43

•885

-87

•90

•03

Atmospheric isentropic analysis requires an investigation of conditions on a sur-
face of constant entropy. In the atmosphere, provided there is no condensation, these
surfaces are identical with surfaces of constant potential temperature and also with
surfaces of constant potential density. For oceanic water the relationships between
entropy, potential temperature and potential density are not so straightforward as
for atmospheric air and in particular, under normal conditions the surfaces of con-
stant entropy, constant potential temperature and constant potential density in the
sea are not identical sets of surfaces. It can easily be understood that especially the
surfaces of equal potential temperature are not identical with surfaces of equal po-
tential density by considering the complete dependence of the latter on the locally
varying salinity which plays only a minor role in the calculation of the potential
temperature. Thus for an investigation of the spreading of the water masses neither
one of these surfaces can be favoured, since each satisfies certain conditions which
seem to be necessary for such considerations, but are not sufficient to give any of the
two methods a special preference. It is thus equally incorrect to denote the method
of using surfaces of equal potential density as reference surfaces as "isentropic"
method because they have nothing to do with entropy which for sea-water is difficult
to define.

Since there is, as previously pointed out, very little'difference between the potential
density ae and the density Cf (down to a pressure of 1000 decibars or a depth of 1000
m), instead of strictly "isentropic analysis" simply the distribution of the oceano-
graphic factors on surfaces of constant a^ has been studied. The method is thus quite
simple in practice, but its usefulness is rather limited if one considers strictly its proper
limits of applicability, and it offers little advantage over the "core layer" method and
other similar methods which will be discussed later. The displacement of water masses

194 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution
60° 50° 40° 30° 20° 10°

Fig. 90. Salinity distribution on the 25-5 o^-surface in the north-east Atlantic between 0° and
about 30° N. (according to Montgomery) (only decimals have been entered as salinity

values).

within such an isopycnic surface must by definition proceed without changes in the
potential density and thus without changes in the potential temperature and the
salinity (or in the oxygen content also). If the distribution of the temperature and
the salinity (or of the oxygen content) pictured on such a surface show signs of
change, these must be due to mixing, and it is therefore possible to investigate these
more closely and to follow the main direction of flow and the spreading of different
water types by means of isolines.

Thereby it was assumed that the mixing in such "isentropic" surfaces occurs pre-
dominantly in horizontal direction (that is in the direction of the surface) and to a
much smaller extent in vertical direction (normal to the surface). This assumption is
not entirely justifiable and may be satisfied only in cases where the Cj-surface runs
just within the density tran ition layer, since here the exchange coefficient in vertical
direction is strongly reduced due to the great stability of the vertical stratification,
and lateral exchange is thus very much favoured. Outside the density transition layer,
however, there is no reason to assume that the effect of vertical mixing is less important
than that of lateral mixing, especially as the reduced magnitude of the vertical ex-
change coefficient is compensated for by rather pronounced vertical gradients of the
oceanographic factors, as was seen earlier.

Montgomery (1938) has applied this method to determine the oceanic circulation of
the upper layers of the southern North Atlantic. The results of this investigation will
be discussed later in connection with the dynamics of ocean currents; here only the
method for the use of the o-^-chart will be presented. Figure 90 gives an example of such

Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 195

an "isentropic" chart for the salinity distribution at the 25-5 c7<-surface in the North
Atlantic between 0° and 30° N. This surface intersects the sea surface at the dotted
line and south of this it lies mostly at a depth of between 75 and 125 m. The arrows
show the main direction of spreading of the highly saline {S) and low saline (F) water
according to Montgomery. The arrows pointing in the west-east-direction show the
Equatorial Countercurrent and correspond to actual flow. Only the east-west-arrows
in the low-salinity tongue between the Equatorial Current and the southern branch
of the north Equatorial Current and those directed from south to north off the West
African coast may have little relation to actual currents; the first low-salinity tongue
represents the salinity minimum between the intrusions of highly saline water to the
north and the south, the latter minima are due to upwelling water off the West
African coast.

5. The Vertical Equilibrium in the Ocean and Stability

The use of the potential temperature 6, or the potential density a^, as criteria for the
equilibrium conditions in the sea is only correct if the salinity is constant everywhere.
Under these conditions the equilibrium is stable, indifferent (neutral) or unstable
according to whether daejdz = 0. Correct equilibrium conditions can be derived in
the following way: a small mass of water displaced from a level r by a vertical distance
A^ towards the surface comes to a density p, while the surrounding water at this point
has a density p'. This displaced water quantum will then be subject to a vertical accelera-
tion proportional to p — p'. If the difference is positive then the displaced water mass
will be subject to a downward force tending to move it back to its previous position ;
the equilibirium is then said to be stable; if the difference is negative then it is subject
to an upward force tending to displace it further and further away from its new
position — the equilibrium is then unstable. If, after a displacement, it always has the
same density as the surrounding water then the equilibrium is indifferent (neutral).
The difference p — p' per unit length is thus a measure of the state of equilibrium.
Hesselberg (1918) therefore denoted the expression E = Spjdz as "stabihty", where
Spjdz is the individual change in density (in contrast to dp/dz which gives the geo-
metric change in p with height). For positive values of £ the stratification is stable and
is not altered by vertical displacement of individual small water quanta. For negative
values of E the stratification is unstable and the slightest disturbance is sufficient to
cause a new adjustment in stratification (Ekman, 1920). Between layers with positive
and negative stability there is always a surface with E — 0. A small mass of water on
displacement to the side where E is positive is always driven back to the surface, but
a displacement to the side where E is negative removes it more and more from that
sui face.

Hesselberg and Sverdrup (1914, see also, Schulz, 1917) have given a simple
method for the calculation of the quantity E. If a small water quantum at a depth z
at point a (Fig. 91) is subject to a pressure p and has a salinity s and a temperature ^,
at a depth z + dz, the corresponding values are p -\- dp, s ~\- ds and {}• + d^. If the
water quantum is displaced near to point a, it will be subject to the pressure p and it
will retain a salinity s + ds, but its temperature will change due to adiabatic expansion

196 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution

>a • b

s+ds

^+d4-dt

z-¥dz
p + dp
s+ds

4+di

Fig. 91. Calculation of stability.

by dr so that its temperature becomes & + d&
points b and a will thus be

dr. The density difference between

dp dp

Pp,s+ds,&+d»-dT — Pp,s,9 ~ ^ ^^ ^ 'M. ^^^ ~ ^'^^■

The stability E is then given by the expression :

dp ds dp id'd'
ds dz dd' \dz

E =

dr
Jz

The geometrical changes in salinity and temperature ds\dz and ddjdz for the depth z
at a give station can thus be determined from the given values of T and S, and the
temperature gradient drjdz as well as dpjds and dpjdd' can be found from hydro-
graphic tables.
If the salinity is constant in vertical direction (dsjdz = 0) then

dr
d

doe
1z

This is in agreement with the previously given equilibrium condition for the potential
temperature. For a given vertical change in salinity its effect on E is so large that it
cannot be ignored.

"Meteor" St. 310 (see Table 79) has been selected as an example for the vertical
stability distribution; the E distribution is given in Table 80.

In the top layer down to 25 m there is a very weak negative stability and just below
the top layer E lises to very large values. This is the density transition layer where the
stratification of the water is extremely stable. Underneath the stability decreases some-
what to assume a value of about 100 at the boundary between the oceanic troposphere
and the stratosphere. Tt then decreases steadily approaching neutral equilibrium in the
greatest ocean depths. All tropical and subtropical stations show similar conditions.
Towards polar latitudes the large positive values of £" in the upper layers disappear and
are replaced by a more uniform, however, not espec'a!ly la ge stability; only the sur-
face layer can be disturbed to any extent by changes from season to season.

The vertical stability at great depths in the deep-sea trenches is of particular interest.
Since in these the salinity is very largely constant the vertical stability conditions can
be estimated fairly accurately from the potential temperature (see p. 127). According

Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 197
Table 80. Stability in the Atlantic (10^ X E)

Depth

"Meteor"

All

Depth

"Meteor"

All

im)

St. 310

"Meteor"
stations

(m)

St. 310

"Meteor"
stations

0-25

-6

900-1000

25-50

1541

—

1000-1200

50-75

360

—

1200-1400

75-100

377

—

1400-1600

100-150

343

—

1600-1800

1800-2000

181

150-200

514

—

200-300

219

202

2000-2250

12-6

300-400

168

151

2250-2500

9-5

104

400-500

152

120

2500-3000

8-6

8-2

3000-3500

7-3

7-9

500-600

113

102

3500-4000

3-6

600-700

108

700-800

4000-4500

—

8-6

800-900

4500-5000

—

3-3

to the observations made by the "Snellius" Expedition (Schubert, 1931) the Philippine
Trench shows the values of £■ X 10^ given in Table 81.

PoLLAK (1954) has given a different definition for the stability which has some ad-
vantages in many cases. It gives somewhat different values for E, but differences re-
main in the limits.

Table 81. Vertical stability (10^ x E) in the Philippine Trench according to
the observations of the ''Snellius''' Expedition

Depth
interval

3500-
4C03

4000-
4500

4500-
5500

5500-
6500

6500-
7500

7500-
8500

8500-
10,030

108 X E

+ 1-2

+0-8

-0-3

+0-5

+0-7

+0-2

+0-8

It may also be of interest to deal with another equation for the vertical stability
which shows clearly the difference of the £-values from the vertical density gradient
datjdz which has often been used previously as a measure of stability. The density
Ps,i^,j, in situ is calculated from hydrographic tables by applying three correction terms
to the value

The first of these e^, depends only on the pressure p, the second e.,,,, depends on the
salinity and the pressure and the third e^.j, depends on temperature and pressure, •
Then

Ps,9,p = 1 + [o'l? + e-j, -f €5, J, -f 6^, J,]

and

Ps^ds, &^d&-dT, p — 1 +

^d+dd- -f ^j) + ^s+ds,p + ^9+dS, p ^ dr

198 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution
From these equations one obtains

hp = da, + -^ds-\- -^^ ^^ - a^ ^^

and thus

dag. dcg^ J, ds de&^ ^ d'& dp dr
^ ^ ~dz '^ 8s dz "^ ~dif dz ~ ~8& dz'

In this expression for E the first term is usually the main one and the others are only
correction terms; the second term shows the effect of changes in salinity, the third
shows the effect of changes in temperature on the compressibility while the fourth
allows for the adiabatic temperature effect. Estimation of the order of magnitude of
these terms shows that they cannot be neglected; the effect of the temperature differ-
ence on the compressibility must already be taken into consideration for depths be-
low 100 m; in deeper layers also the adiabatic effect is of the same order of magni-
tude as the first term. In general only the effect of changes in salinity is mostly small.
The quantity do^jdz for itself thus cannot give a very precise measure of the stability.

6. The Distribution of Stability in the Atlantic Ocean

Schubert (1935) has carried out a detailed examination of stability conditions in the
Atlantic Ocean — in particular of regional stability differences in vertical sections and
on horizontal charts. Table 80 also gives mean values of E for the entire ocean calcu-
lated as means of all "Meteor" stations; the surface layer down to 200 m, i.e. the
zone of disturbance, has been omitted. Of the many irregularities in the vertical distri-
bution at individual stations, only two remain in the mean values, the most important
being that at 1000 m. This is a definite intermediate stabiHty maximum. From the
location of this rather strong interruption, or sometimes even reversal, of the normal
decrease of stability downwards, the decrease in stability is considerably larger than
before. This irregularity is present at about the same depth throughout the total
ocean in temperate and tropical latitudes, and is connected with the subantarctic
intermediate water. Its basic cause is the reversal in the salinity gradient.

There is another secondary maximum imposed on the regular decrease of the E-
values at a depth of 2000-4000 m. In contrast to the more sudden change at 1000 m
a weak and more gradual increase in stability is characteristic.

In the regional variability of the stability in particular, a strong decrease towards
higher latitudes stands out. The higher values of E disappear already beyond 50°
latitude; the greater uniformity and lower values indicate that only in higher latitudes
do favourable conditions for vertical displacements of water exist. Solely by this, higher
latitudes become the principal regions of origin for the deep-sea circulation of the
oceanic stratosphere.

Characteristic stability conditions are found in the top layer down to 100 m or
occasionally to 200 m where frequently negative values occur. Apart from cases in
the upper 25 m, where they are very frequent, these negative stabilities were formerly
regarded as due to observational errors (especially in the salinity). However, variations
of 0-01%o are in fact quite sufficient to explain them (Helland-Hansen, 1910).

Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 199

Observations of more recent expeditions have shown that negative stabilities extend-
ing down at the most to about 250 m are of such a frequent occurrence, that they
are difficult to account for by observational errors alone. For example, in ninety-five
cases with E greater than —100 the observational errors must be 0-04%o in S or 1°C
in temperature. There is, however, further confirmation of the reality of this pheno-
menon. This comes from the occurrence of negative values throughout the entire
layer, and the fact that mostly a pronounced regional distribution of stations with
negative values of E is found which would scarcely be possible if random observa-
tional errors would have been made. In the Atlantic, for example, there is an extended
area with negative values of E in the entire open ocean from 50° S. to 20° N. The
highest negative values (< —200) fall within a latitudinal zone between 15° and 20° S.
and there is probably a corresponding zone also in the North Atlantic approximately
between 20° and 30° N.

This instability in the top layer in tropical and subtropical areas must be due to the
eff"ectiveness of evaporation. The increase in salinity and the decrease of the temper-
ature at the surface leads to an increase in density and to a reduction in stability.
Solely incoming radiation during day time works in the opposite direction, which
compensates the density increase by a corresponding rise in temperature, but during
night time when incoming radiation is missing and evaporation continues, the density
increase will predominate and negative stability values can persist for a considerable
time as long as the intensity of evaporation is sufficient. It is, however, a rather pe-
culiar phenomenon that a vertically unstable stratification can be maintained for a
longer time over such an extended area in the top layer in spite of convection and
mixing.

Fig. 92. Circulation in a convection cell according to Benard.

Perhaps a possible explanation lies in the "convection cells", first observed and
investigated experimentally by Benard (1901). He was able to show that when a rela-
tively thin layer of a liquid with volatile components was cooled by evaporation, the
entire mass of the liquid divided into a number of cells. In each of these the liquid
rises in the centre, diverges in the upper part of the cell and descends again in the
outer parts as shown schematically in Fig. 92. The diameter of the cells corresponds to
about three or four times that of the thickness of the liquid layer. Instability in the

200 Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution

stratification is associated with such convection cells and is maintained by the circula-
tion. Rayleigh (1916) and Jeffreys (1928) investigated sucha Benard cell theoretically
and showed that there could be an equilibrium state with an upper layer of greater
density on top of a lower one with smaller density if the vertical density difference
between the upper and the lower layer was less than a certain limiting value given by
the inequality

Agli" '

where k is the molecular thermal conductivity coefficient, v is the kinetic viscosity co-
efficient and h is the thickness of the liquid layer. The unstable density difference is
largest in the upper part of the layer; as long as the loss of heat by evaporation tends
to maintain the unstable stratification the circulation will continue. It will, however,
cease immediately as soon as the evaporation ceases. If there is a steady current in any
direction in such a liquid the convection cells resolve into long bands with a corres-
ponding transverse circulation.

It is not impossible that the existence and maintenance of density instability in the
top layer of the ocean has something to do with such phenomena. However, in order
to simulate conditions actually found in the ocean, the influence of radiation and
evaporation and especially that of the eddy conductivity and eddy viscosity must be
taken into account in the above inequality, instead of the molecular thermal con-
ductivity and the molecular viscosity. For a layer 25-50 m thick resting on top of a
transition layer with a stable stratification, the above inequality will give a value for
(p' — p) of the order of magnitude of the observed negative stabilities. By the effect
of the circulation a mechanical instability is thus changed into a dynamic stability.

In more recent times the theory of convection cells has been considerably advanced
and has been discussed in detail in a symposium on the problems of boundary layers
and convection cells in the Section of Oceanography and Meteorology of the New York
Academy of Sciences, 1942. Stommel (1947) has presented a summary of the theory
of convection cells which should especially be mentioned. Neumann (1948) has paid
special attention to cell convection in the sea and has shown that indifferent (neutral)
stratification occurs only when

^0 A"-
F = - ^

Pg h''

where A^'is a. dimensionless quantity of the order of M X 10^ in the ocean, A is the
vertical exchange coefficient and h is the thickness of the layer. This equation follows
directly from that given by Rayleigh if the above-mentioned change from molecular
into turbulent conditions is introduced. The greater the thickness of the layer h and the
smaller the exchange coefficient A, the smaller is the decrease in density with depth
that is still compatible with static equilibrium. Convection starts only when denser
water is situated on top of lighter and when A in the above equation exceeds the
critical value 1 100.

At the "Meteor" anchor station 385 (16° 48-3' N., 46° 17-1' W.; second continua-
tion of the German North Atlantic Expedition, February 1938) it was found, as a

Density of Water Masses in Ocean, Vertical and Horizontal Density Distribution 201

mean of sixty series of observations, that the water at the sea surface was always appre-
ciably denser (heavier) than that at 6 m depth and even at 1 5 m depth the water was still
specifically lighter than at the surface. Taking ^ = 100 g cm-^ sec-^ and h = 500 cm,
then E = — 16 x 10~^ ; this means that convection is initiated in this layer at this value
and not at £" = 0. If the turbulence becomes stronger the critical value of E increases
rapidly and strong density gradients are required for any start of convectional motion.
The long lines of foam often observed on the surface of the sea can be regarded as
"convection rolls" formed by a combination of a strong current in a single direction,
and circulations in convection cells in the above sense. Their frequent occurrence is an
indication that regular formations of Benard convection cells occur in the sea.

Chapter VI

The [TS] -relationship and its
Connection with Mixing Processes and
Large Water Masses

1. Temperature as a Function of Salinity and Large Water Masses

Temperature and salinity vary with the depth h or the pressure p, and an investiga-
tion of the vertical distribution of these factors is based mainly on a graphical repre-
sentation of the variation of these quantities with depth h. In this way it is almost
unconsciously assumed that these factors (temperature and salinity) are independent
of each other. This is, however, not the case. Assuming salinity as a function of tem-
perature or plotting it against temperature in a system of co-ordinates (tempera-
ture as ordinate, the salinity as abscissa) the points for each depth are not distributed
at random over the diagram but fall on a definite, more or less smooth curve. It is
found that for oceanic regions with uniform oceanographic and special climatic, as
well as undisturbed flow conditions, the [r^J-relationship is quite characteristic. A
given temperature corresponds to a given salinity regardless of the depth. The prac-
tical significance of this [r^J-relationship was first pointed out by Helland-Hansen
(1918) and since then it has become increasingly important. Any given water type, a
water mass, formed continuously in a particular oceanic area for any kind of condi-
tions is characterized by a definite temperature and a definite salinity. If this water
mass is homogeneous then the oceanographic factors in it are constant and it can be
represented on a [r5]-diagiam by a single point. If this water mass is moved in any
direction without altering its physical-chemical structure the point does not change its
position on the diagram. However, under influence of certain processes, for instance
mixing, radiation or evaporation, the water mass loses its homogeneity and the
position of the point in the co-ordinate system is changed. Such changes occur espe-
cially in the top layer (down to 200 m), where climatic conditions are able to pro-
duce continuous "disturbances" in the normal state. Beneath the top layer with dis-
turbances, however, conditions in the ocean are qudL^i-stationary and thus every station
has its characteristic [r5']-curve which for that special station remains largely in-
variable. This constancy is, however, not only true for each individual station but
applies also in a somewhat wider sense to more or less larger oceanic spaces. Standard
curves can thus be constructed for diff"erent regions and conclusions can be drawn
about the origin and spreading of a water mass from the deviations of the values at a
particular station from those of the standard curves.

202

[TS]-relationship and Connection with Mixing Processes and Large Water Masses 203

Figure 93 shows an example of such a [rS']-curvefor "Meteor" station 171 in the cen-
tral part of the South Atlantic. Its shape is characteristic for the entire South Atlantic
from 40° S. to beyond 10° N. Its constancy over such a large area expresses well the
strong conservatism of vertical stratification which is of course necessary under sta-
tionary conditions. If, in addition, lines of equal density Cf (isopycnals) are also included
in the same diagram, as was done in Fig. 93, a rather instructive although not com-
pletely correct representation of the stability of vertical stratification is obtained. If

34-2

350

360

370

Fig. 93. [75] -curve for "Meteor" St. 171 (22° 1-5' S. 23° 470'W.) in the central part of the
South Atlantic (the thin dashed curves are the isopycnals Of).

the [TS]-cmyQ of a certain layer runs approximately parallel to the isopycnals the
stability in the layer is only small but if the [rSJ-curve cuts the isopycnals at a wide
angle the stability is larger. For greater accuracy the [J'5']-curve must be constructed
by using potential temperatures, but the differences in most cases remain small.

As with temperature, so can any other property of sea-water be combined with the
salinity in exactly the same way. Such a combination was made in particular with the
oxygen content in order to see how changes in the oxygen content affect the temperature
and salinity conditions, which determine the water mass.

2. Practical Significance of the [T^S"] -curve

The [rS'l-curve offers advantages in the scientific preparation of oceanographic data
and is used to detect errors and to make it homogeneous. If the value for a particular
depth at an oceanographic station does not fall on the simple, regular and usually
smooth [rSJ-curve it can be confidently assumed that there is an observational error
or a fault in calculation (for examples see Merz, 1925). The [r5']-curve is thus a
reliable criterion of the accuracy and homogeneity of a set of data. Since curves for
neighbouring stations are similar all values can be checked immediately, but a faulty

©>

204 [TS]-relationship and Connection with Mixing Processes and Large Water Masses

observation can also thereby be replaced by an approximate, rather more correct
value. Only in this way is it possible to perform an objective and satisfactory "inter-
polation" of oceanographic values in order to fill gaps (missing data) in the observa-
tional material.

3. The [r5'] -curve and the Mixing of Water Masses

If two homogeneous water masses are mixed in any given proportion, the mixture
will have a definite [rSJ-curve. Each of the two homogeneous water masses is
characterized by the two points, 1 {s^, §i) and 2 {s2, i dz), in the co-ordinate system,
proceeds in the ordinary way; if two masses are mixed in the ratio nti : Wg then the
mixing final temperature and salinity of the mixture will be given by

-& =

mi + /«2

s =

m^Si + AWa^a

An example is presented in Fig. 94 where a homogeneous water mass U (10°, 35%o)
from 100 m to 500 m depth is situated above a second mass Z (5°, 34-5%o) which extends
down to a depth of 900 m (Defant and WiJST, 1930). These two homogeneous water
masses are represented in the [7'5']-diagram by the two points U and Z. The boundary
surface at 500 m depth, which is initially a sharp physical discontinuity surface,
gradually disappears due to mixing. Different stages of this destruction of the dis-
continuity is shown on the left-hand side of Fig. 94 (Defant, 1929). It is obvious that,
whatever the ratio of mixing of the two water masses may be, the mixture will be
represented on the diagram only by points lying between U and Z. However, the
graphical construction shows that all points of the mixture must be situated on the
straight line from U to Z and that only the depth changes on this line according to the
intensity of mixing. This is readily shown theoretically (Defant, 1935). It can also
be demonstrated that the distance of any point along the straight line from the two
end-points (representing the two original water types) is inversely proportional to the
ratio of mixing, the result of which is the mixed water type at the point in question.
It is thus simple to determine from the position of a point relative to the end-points
U and Z in Fig. 94 to what degree (in percentage) the final mixed water mass under
consideration is composed of each of the original water types.

The case where three water types are mixed is illustrated in Fig. 95. The three types
are:

Water mass

U Z

Layer thickness (m)

100-500 500-1 COO

1000-1500

Temperature (°C)
Salinity (%„)

10
350

5
34-5

5
350

The thermal boundary surface at 500 m disappears in the same way as in the pre-
vious case. The salinity boundary surface does the same up to the time when the inter-
mediate water mass Z becomes involved in its total height in the mixing process and in
that way is slowly destroyed at its core. An advanced stage of this is shown in the

[TS]-relationship and Connection with Mixit^g Processes and Large Water Masses 205

Mixing

of 2 homogeneous woter bodies

200

10°

400

8°
6°

600

r:^'-

- z/

600

S- f{t)

1000

1 1 1 1 1 1 1 1

2°

1 1 1 1 1 1 1 1

4° 6° 8° 10°

34.4 346 348 35 0
S, 7oo

34 4 34 6 34 8 35 0
5, %o

200

10°

400

■

8°

6°

- z4

600

1 /

( ■■'■■■'

800

;

[■

4°

S--f[t)

1000

1 1 1 1 1 1 1 1

1 1 I 1 1 1 1 1

2°

1 1 1 1 1 1 1 '

4° 6° 8° 10°

34 4 34 6 34 8 35 0

/, "C

5, %o

S. 7oo

Fig. 94. Mixing of two homogeneous water masses and the resulting [JSJ-relationship.

Mixing of 3 homogeneous water bodies

500- 2

1000

1500

A-::-^^

y-'j:. -^)

v^'"

z
r

'S^i^.

r
1 1

1 1 1 1

1 1 1 ! 1 1 1

T
1

4 6 8 10

/, "C

34 4 34 6 34 8 35 0
S, 7o,

34 4 346 348 350
S--f\t)

Fig. 95. Mixing of three homogeneous water masses.

206 [TS]-relationship and Connection with Mixing Processes and Large Water Masses

^'-distribution by the dotted line. In the [J^J-diagram the mixing of the three water
types is represented by the broken hne UTZ, but the point Z remains on this [r^]-
curve only as long as the core of the intermediate water is not involved in mixing with
the water masses U and T. When this happens the Z-point moves into the acute angle
and the [TS]-c\xr\c no longer has a peak point at Z but becomes rounded there.
The reversal point and the concentration of the depth marks around it shows the core
of the intermediate water already affected by mixing, but even at this advanced stage
the mixing of the three water masses can be represented by a curve VZT made up of
two straight lines.

Analysis of actual [r5']-curves of the oceans show essentially these main theoretical
characteristics; they are remarkably constant over large oceanic regions, they have
characteristic reversal points associated with the cores of the individual water types
and large parts of these curves often show a surprising approach to a straight line.
In these cases the [r5']-curves allow a precise determination of the depth, temperature
and salinity of the water masses, which finally combine and form individual water
types in the deep layers and they also allow the percentage of the individual compo-
nents to be found at all intermediate stages.

In the foregoing discussion it has so far been assumed only that mixing proceeds
according to the usual mixing rules ; the magnitude of the exchange coefficients is not
involved. The percentages of the original component-waters before mixing give no
information on this point. For obtaining a connection with the [r^j-curve the basic
equation given on p. 106

d^s 8s

is required. This implies that in order to secure a stationary state the vertical exchange
and the horizontal advection must completely balance. By choosing for the origin
of the A"-coordinate (pointing positively in the direction of movement of the water
mass under consideration in a longitudinal section) that point where the water mass 1
is still pure, then the salinity at a distance x will be s^ and at a distance x -{- dx will
be Sx+Ax and one obtains with sufficient accuracy:

8s Az 8^s

— Ax = s^-\ ^-^

8x pu 8z'

^x-^-Ax — ^x \ a^^-^ ^x y' p_2 ^•^•

If Sj. is formed from s^ and S2 in the proportion m^ and mg of water masses 1 and 2
and Sj._^^x in the proportion m^ — Am and Wg + ^w, then using the mixing rule the
above equation transforms to

(52 — Si)Am Ag 8^s .
nil + f^h P" ^^

If now m^ + Wg is replaced by the distance D of points 1 and 2 on the [r5']-curve and
Am by the distance AD of the point (s -{- As; >& -\- Ad) from the point (5, d), then

A 2 S2 — Si AD 1

pu D Ax (8^sl8z^)

[TS]-relationship and Connection with Mixing Processes and Large Water Masses 207

This formula allows the value at any point along the line of spreading of a water type
to be calculated from the [^l-curve if the vertical distribution of salinity (or of
temperature) is known. In the special case of a tongue-like spreading s is given with
sufficient accuracy by the simple form (see p. 106, et seq.)

•s = -^0 + Rx) cos ^^ z.

Then

and for the core layer (z = 0)

8'^s _ 772

dh 7r2

Because

/(I) = 5o — 5i and /(2) = ^o — Sz,

and therefore

S2-s^=f{l)-f(2)

we obtain

±,_^ /(I) -/(2) ^J>

pu TT^D fix) Ax '

The application of this equation to the core layer of the subantarctic intermediate
water along the western section in the Atlantic gives values for between 0-6 and M
which is in rather good agreement with those determined by other methods. However,
this method, using the [75]- relationship, also gives only the ratio between vertical
exchange and velocity.

An interesting method that also uses the [r^l-relationship and allows a deeper in-
sight into the process of mixing has been given by Jacobsen (1927). Consider a vertical
column of water with cross-section of 1 cm^. From this column we consider two
cubes (volume 1 cm^) ai A [z = 0) and also at a point J5 at a distance z beneath A.
In the course of a mixing process, which should follow the laws valid for diffusion and
occurs within the total column which we assume at rest, there will be an exchange of
q cm^ of water in the time of / sec between the two cubes. If the displacement of the
water quanta during the mixing process follows a Maxwellian distribution then

q = ke-°-'^\

Since there is no increase in mass in the entire water column the integral of qdz from
— 00 to +00 must be equal to 1 , This gives a^ = nk^. The amount of salt in cube B
is ps X 10~^, where the salinity s is given in per thousand and the increase in salt
amount in a small time dt according to the exchange equation is

Corresponding relationships with Sq and Asq applies to cube A. The sahnity (sq +
Asq) in the cube after a time t is the sum of the salt amount originally present and the

208 [TS]-relationship and Connection with Mixing Processes and Large Water Masses
salt increase due to the exchange of water quanta by mixing and is thus

I+cc
ps X 10"^ qds.

Putting with sufficient accuracy

(ds\ 1 [d^s\

^dzj Q 2 ydz^/Q
and using the above expression for q gives for the point A (z = 0)

Then k is expressed in terms of the exchange coefficient A. The [r^j-relationship for
the water column under consideration is presented in Fig. 96, At the reversal point
/i,z = Oandthedepthmarks +1, +2, +3, . . ,,and — 1, — 2, — 3, . . ., respectively,
correspond to the centre points of water cubes of 1 cm^; the cube at A is thus the zero
cube. The circle with a radius R (AO = R) approximates closely to the [r5']-curve
at the point A. For the part of the curve under consideration the depth marks are so
situated that the arc between each pair of depth marks always corresponds to the
same angle a in point O. It is necessary to find the co-ordinates (temperature and
salinity) after / sec of the zero cube initially at A . Two points M and A^ at vertical
distances z and z -\- dz cut out a volume element of dz cm^ of water. According to
the previous discussion a quantity of water gdz is transferred from this element to
the zero cube in t seconds. The same quantity of water gdz is also transferred from the
symmetrically situated volume element M'N' in the same time. These two quantities
of water mix, and according to the mixing rule the mixture 2qdz corresponds on the
[rS'j-diagram to the small interval BC which is situated on the radius of curvature AO.
It is determined by the distance

AB ^ r = R - Rcos (za) = ^Rah^
and

BC = dr = la'^z'^dR.

The water masses entering the zero cube during time / are not only transferred from
the two cubes MN and M'N', but also from all other cubes above and below, and it is
easily understood that the T and S values for all these water masses must lie on the
radius AO. Mixing of all these differential quantities gives the co-ordinates of the zero
cube after / sec. Its position on the [rS'j-diagram will be fixsd by the distance Z
along AO. According to the mixing rule this must be given by

2q dr dz.

One therefore obtains

Z = - RaH.

[TS]-relationship and Connection with Mixing Processes and Large Water Af asses 209

On the other hand, the chord drawn through Z perpendicular io AO intercepts an arc
on the [r5]-curve with a centre angle ha (the depth marks at the end-points of the
chord are -\-\h and —\h) and

Z = R- RCOS ajla) + i RaW.

Comparison of the two values for Z finally leads to an exchange coefficient

This equation can be used for the numerical determination of A if the [r5^-curve
for a water column has been found by observation for successive times. In Fig. 97 I

Fig. 96. Calculation of exchange coefficients by the method of Jacobsen.

denotes the initial distribution which is followed by distribution II after / seconds.
It shows the changes that have taken place in the water column during time t. The
points are depth marks for the determination of h. The tangent at A cuts the [r^]-
curve I at the depth marks h^ and //g; the size of ^ is thus /?! — h^. The equation then
allows calculation of the exchange coefficient yi if Ms known.

The Jacobsen method appUes almost only to oceanic regions which are practically
motionless and in which the gradual disappearance of a disturbance in the vertical
structure due to vertical mixing can be determined by successive measurements. An
application to stationary water displacements is possible using the principle
that phenomena occurring one after the other in time can be replaced by others
occurring side by side in space. Then the [r^J-diagrams I and II in Fig. 97 represent two
successive stations at a distance L in the direction of water displacement. If u is the
velocity of this displacement then L = ut and from the above relation one obtains
Ajpu =h ^jSL. It can be seen that this method again gives only the ratio Aju.

210 \TS]-relationship and Connection with Mixing Processes and Large Water Masses

Fig. 97. [rSJ-relationship in a water column at successive times.

4. Further Examples of the [J'S'] -Relationship

Extensive use has been made of the [rS'] -relationship in oceanographic investiga-
tions of different oceanic regions. A detailed discussion of these investigations belongs
to the individual sections on special oceanography and would be out of place here.
The attention of the reader will therefore at present be directed more to the method
used rather than to the phenomena characteristic for different parts of the ocean.

A most intensive analysis of the [TlSj-curves for a single ocean was first made by
Jacobsen (1929) on the data collected by the "Dana" Expedition. He divided the
North Atlantic into twenty-four areas with approximately uniform conditions, and
he derived mean characteristic [r^SJ-curves for these areas, using then these curves
to give an interpretation of the formation of the stratification by mixing of the five
principal water types. A homogeneous set of data for the preparation of [T^J-curves
for the South Atlantic as far as 10° N. has been provided by the "Meteor" Expedition.
Figure 98 presents [rSj-curves for the West Atlantic Trough as an example for meri-
dional changes. In this region extending over more than 44° of latitude (almost
5000 km) the thermo-haline structure follows the same law almost without exceptions.
It is in principle fixed by five water masses U, Z, T, Bn and Bg and corresponding
mixing curves. Basic values are given in Table 82.

Five points on the diagram characterize each of these water masses together with
straight lines joining them, on which the mixed water masses must lie. The variations
of the actual [r^j-curves from these ideal curves of pure mixing are surprisingly small,
especially when there is a sufficient mass of water in the cores. This is usually the case,
though for the subantarctic intermediate water as it progresses from south to north
the [7'6']-curve moves farther and farther into the angle between VZ and ZT, as is
required by theory, showing that in this comparatively thin layer of water the core is
also involved in the mixing process. This case can be used to calculate the ratio
Ajpu for the spreading of the sub-antarctic intermediate water by applying the
Jacobsen method (Defant, 1954). Figure 99 shows [r-SJ-curves at four successive
oceanographic stations from south to north in the Western Trough of the South

[TS]-relationship and Connection with Mixing Processes and Large Water Masses 211
Table 82. Water masses of the South Atlantic between 33° S. and 11° N.

Temp.

Salinity

(%o)

Antarctic components

Subantarctic intermediate
water

3-25

3415

Antarctic bottom water

0-4

34-67

North Atlantic components <

North Atlantic deep water

3500

North Atlantic bottom water

2-5

34-90

Beneath the disturbed top layerJ
approx. 100-200 m \

Subtropical lower water

180

35-93

Atlantic but solely for 400-1400 m depth. The values for L and h in the Jacobsen equa-
tion on p. 210 can be obtained immediately from the curves and in that way Table
82fl is obtained.

The mean value of Ajpu is 0-74 and for ti = 10 cm/sec the quantity A is 7-4 cm^/sec
which is in good agreement with values determined by use of other methods. A

2°° : West Atlantic Irouqh ^ y

S=f(t) J.^ >^ ^'^

340 «» 3&0

340 14 5 3S0

340 M5 350

34-0 M-5 350

°f .'^ ™®^' J 'C '"• '"'^v -^ i-<'''''5''' /'^!c i[^' /ihU's^'^T

i°t ■ Bs^ Bsa*- Bsv<c . . Bsy Bs«<5

34 0 5-15 35 0

Fig. 98. [rSJ-curves for a series of stations along the Western Trough of the Atlantic.

Table 82a. Determination of Aj pii by
means of the Jacobsen method

Station pair

Ai pu

(St. no.)

(km)

(m)

(cm)

160-202

600

215

0-96

160-297

3050

380

0-59

160-290

4150

485

0-71

202-297

2450

350

0-625

202-290

3550

510

0-92

297-290

1100

220

0-55

212 [TS]-relationship and Connection with Mixing Processes and Large Water Masses

considerably more detailed treatment of problems briefly outlined here has been given
by WiJST (1936) using this data and applying the so-called "core-layer method". The
characteristic properties of a water mass are retained in the core layer and an analysis
of changes in the core layers is therefore of decisive importance for an investigation
of the spreading of a water type. The most difficult and essential part of such an in-
vestigation is the accurate determination of the core layer of each water type at each
station from the vertical distribution of the different oceanic factors. A single factor
will not necessarily be best for the characterization of the core layer. Thus salinity was
found to be the most suitable indicator for the subantarctic intermediate water and
also for the upper North Atlantic deep water which gets in the North Atlantic a
continuous supply from mediterranean water, while the potential temperature is used
for antarctic bottom water and oxygen content for Lower Deep Water (intermediate
maxima in oxygen content). The spreading of a water type can be found by following
the appropriate indicator. The subantarctic Intermediate Water can be taken as an
example, to show the use of the "core layer method"; here the intermediate salinity
minimum between 50° S. and 20° N. is an excellent indicator. The depth of this
minimum and its salinity, temperature and oxygen content can be evaluated from the
vertical curves of all stations. This water type sinks as shown by the analysis at the
southern oceanic Polar front. The 100 m depth line runs parallel to and immediately
to the north of it, and from here the depth of the core layer is shown by the isobath
on Fig. 100. From 45° S. to 39° S. the core lowers continuously and rapidly from 100 m
down to 800 m and reaches its greatest depth on the average at about 900 m between
37° S and 30° S. It then rises to about 800 m, falling again to 900 m or more north
of 10° N. A chart of the S distribution shows that rapid lowering of the core goes
parallel with a rapid rise in sahnity from 33-9%o to 34-2%o ; beyond this region the meri-

3V3%o

35-0%<

Fig. 99. [r5]-curves for four "Meteor" stations along a western longitudinal section in the
Atlantic in the area of the subantarctic intermediate water (the figures give depths in 100 m-

units).

[TS]-relationship and Connection with Mixing Processes and Large Water Masses 213

dional increase in salinity is less, reaching 34-5%o at the equator, and from there to
20° N. the increase is again more rapid (up to 34-9%o). According to the distribution
of the isohalines, stronger mixing occurs at the western edge of the ocean. [TS\-
relations of the core layer for the western and eastern parts of the ocean are approxi-
mately the same for the two halves of the ocean so that a standard curve can be pre-
pared for the entire area where this water type (Fig. 101) is found. The point Zp
represents the mean properties of the unadulterated water in the area of formation
near the oceanic Polar front and the last traces of this water type are found at point
O. Dividing the interval between Zp and O into 100 parts allows the determination of

60° 50" 40" 30° 20° « iO° 0" 10'

\/y 90° 80° 70° 60° 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50° £

Fig. 100. Salinity C/oq) and depth (metres) of the core layer of the subantarctic intermediate
water in the Atlantic (according to Wiist).

214 [TS]-relationship and Connection with Mixing Processes and Large Water Masses

3&0

tX2

Fig. 101. Standard curve of the [TSJ-relationship for the entire area of the core layer of the
subantarctic intermediate water in the Atlantic.

the percentage influence of each individual component at any point in the spreading
area. From these percentages for all stations a chart can be made with lines of equal
percentages of subantarctic water to give a quantitative representation of the spread-
ing, the gradual mixing and the final disappearance of this type of water (Fig. 102).
Between 45° and 40° S. the subantarctic component of the intermediate water is still
about 7% ; the 50% line runs from South Africa to Cape San Roque and the 40%
line extends from here in a narrow tongue along the continental slope to 9° N. The
principal direction of spread is shown by arrows on Fig. 102.

Wiist has investigated in a similar way the other water types important in the
thermo-haline structure of the Atlantic, and by making a numerical estimate of their
spread from the [rS'J-relations in the core layers. He thereby obtained a quantitative
measure of the mixing of these water types.

A further example of the use of the [rSJ-diagram in the study of the spreading of a
water type is the equivalent thickness-method of Jacobsen (1943); he applied this
method in practice to investigate the penetration of Atlantic water through the
Faroes-Shetland gap into the Norwegian Sea, the northern part of the North Sea
and into the Barents Sea. This water type is characterized by values d' = 10-2°C
and S = 35-45%o. The water mass at any point in this area is formed by a mixture
of pure Atlantic water with other water types. If it were possible to separate pure At-
lantic water from the other types it would have at any point a definite thickness which
Jacobsen called the "equivalent thickness" of Atlantic water. At any station this
thickness can be found from the [rSJ-relation in the following way: Fig. 103 shows the
[r5]-diagram at a "Hjort" station on 1 May, 1935 at 630° N., 3-8° E. The [TS]-
curve approximates the observed points rather closely.

The dashed straight line connects the points of the mixing components A (10-2°C,
34-5%) and P (2-5°C, 34-90%o). The distance AP is divided into ten equal intervals
and indicates the individual share of the two components. The section of the curve

{TS]-relationship and Connection with Mixing Processes and Large Water Masses 215

Fig 102 Spreading of the subantarctic intermediate water represented by lines of equal
percentage content of this water type (according to Wust). The full arrows indicate the main
course of the water spreading and the dashed arrows indicate the (more turbulent) side

branches of it.

216 [TS]-relationship and Connection with Mixing Processes and Large Water Masses

1 1 1 1

1 1 1 1
-SA

■

3><

/€'^ :

- /

"*</

eS^OO'.SN. 3*>46'E.

V. 1. 1935

•5

I .

.III

35rOO

35-50 %o

Fig. 103. Calculation of the equivalent thickness of Atlantic water typeaccording to Jacobsen
for the "J. Hjorf'-Station 1 May 1935 (63° 0' N., 3° 08' E.).

50-100 m follows the straight line AP rather well and therefore shows that the water
masses of this layer are composed principally of these two components. For different
depths the participation of the Atlantic water in the vertical stratification of the ocean
at this station can directly be read from the diagram. The following values are ob-
tained :

Depth in m

150

200

250

300

400

410

Participation of
component A

0-73

0-67

0-60

0-57

0-43

004

001

The equivalent thickness of Atlantic Ocean water at this station is then computed in
the following way :

KO-73 X 50 + 0-70 X 100 + 0-64 x 50 + 0-58 x 50 + 0-50 X 50 + 0-24 x 100

+ 0-02 X 10) = 108 m

The resulting thickness is a measure of the amount of Atlantic water which partici-
pated in the formation of the water column at this station. A geographical distribution
of the equivalent thicknesses over the entire spreading region of Atlantic water in the
Norwegian Sea and in the Barents Sea is a rather good representation of the effect
of the Atlantic current and of the heat carried by this current towards the north,
and also allows a quantitative evaluation.

5. The Water Masses of the Oceans

An accurate analysis of the [rS'J-relation in different parts of the oceans leads to a
closer classification of the water types of which the ocean is made up. By a somewhat

[TS]-relationship and Connection with Mixing Processes and Large Water Masses 217

schematic treatment it is possible to derive from the complicated forms of the [TS]-
relations graphical representations of the characteristic water types of each ocean
that give a good insight into the thermal and haline structure of the sea. This also
assists in a clarification of the formation, spreading and mixing of the individual w^ater
types and thus facilitates a quantitative description of the oceanic circulation of the
deep and bottom layers. Table 83 summarizes the characteristic water types of the
three oceans and indicates the temperature and salinity ranges in them. These limiting
values for temperature and salinity must naturally only be looked upon as a rather
crude measure to demonstrate in what extreme limits oceanographic factors may vary.

Table 83. Water masses of the Atlantic Ocean

Salinity

North Atlantic

Temp. (°C)

(%o)

South Atlantic

Temp. (°C)

(%o)

1 . North Polar water

-1 to +2

34-9

1. South Atlantic cen-

2. Suba ctic water

+ 3 to +5

34-7-34-9

tral water

+5 to +16

34-3-35-6

3. North Atlantic cen-

2. Antarctic inter-

tral water

+4 to +17

35-1-36-2

mediate water

+ 3 to +5

341-34-6

4. North Atlantic deep

3. Subantarctic water

+ 3 to +9

33-8-34-5

water

+ 3 to +4

34-9-350

4. Antarctic circum-

5. North Atlantic

polar water

+0-5to+2-5

34-7-34-8

bottom water

+ 1 to +3

34.8-34-9

5. South Atlantic deep

6. Mediterranean

and bottom water

0 to +2

34-5-34-9

water

+ 6 to +10

35-3-36-4

6. Antarctic bottom

water

-0-4

34-66

Water masses of the Indian Ocean

Temp. CO

Salinity (%„)

1. Equatorial water

4-16

34-8-35-2

2. Indian central water

6-15

34-5-35-4

3. Antarctic intermediate

water

2-6

34-4-34-7

4. Subantarctic water

2-8

34-1-34-6

5. Indian Ocean deep and

antarctic circumpolar

water

0-5-2

34-7-34-75

6. Red Sea water

35-5

Water

masses of the Pacific Ocean

Temp.

Salinity

Temp.

Salinity

North Pacific

(°C)

(%o)

South Pacific

(°C)

(%o)

1 . Subarctic water

2-10

33-5-34-4

1 . Eastern South

2. Pacific equatorial

Pacific water

9-16

34-3-35-1

water

6-16

34-5-35-2

2. Western South

3. Eastern North Pacific

Pacific water

7-16

34-5-35-5

water

10-16

34-0-34-6

3. Antarctic Interme-

4. Western North

diate water

4-7

34-3-34-5

Pacific water

7-16

34-1-34-6

4. Subantarctic water

3-7

34-1-34-6

5. Arctic Intermediate

5. Pacific deep water

water

6-10

34-0-34-1

and A.ntarctic cir-

6. Pacific deep water

cumpolar water

C-l)-3

34-6-34-7

and Arctic cir-

cumpolar water

(-l)-3

34-6-34-7

218 [TS]-relationship and Connection with Mixing Processes and Large Water Masses

The central water at 500-800 m in each of the three oceans forms the principal mass
which always has a structure with an almost linear [r^SJ-relation and thus manifests
its normal mixing in both horizontal and vertical directions. Underneath, and sepa-
rated from it by the Antarctic inteiTnediate water, the deep and bottom waters are
found in the Southern Hemisphere which have a remarkably similar structure in all
three oceans. In the Northern Hemisphere the Atlantic is blocked oif from the Arctic
Sea and has little or no Arctic intermediate water, but the Pacific Ocean, on the
other hand, definitely shows this water and thus this ocean is of a more symmetrical
structure. The North Atlantic and the Indian Ocean show a strongly increased salinity
in the layers between 800 and 2000 m due to the inflow of warm saline water from the
Mediterranean and the Red Sea. These eff'ects are quite strong and are evidenced even
in the southern parts of these oceans. There are no corresponding disturbances in the
Pacific Ocean. On careful examination of Table 83 one cannot fail to regard the
striking similarity of the thermo-haline structure of the oceans with astonishment.
There can be no doubt that this is a consequence of an analogous oceanic circulation
driven and maintained by the same forces.

Chapter VII

Evaporation from the Surface of the

Sea and the Water Budget of

the Earth

1. Introduction

One of the most important problems with which both meteorology and oceano-
graphy is concerned is the water budget of the Earth. It can be assumed with a very
considerable degree of probability that the cycle through which the water passes is
closed. This follows, if a sufficiently long period is taken into consideration, from the
constancy of the amount of water on the earth and from the absence of processes
which could alter, and especially decrease, the total amount of water present. A
stationary water cycle requires that the amount of water passing through any par-
ticular part of this cycle (as either liquid, solid or vapour) should not vary with time,
and particularly that the amount of water entering the cycle by evaporation from the
ocean is returned to it. In that way there is never any permanent gain or loss of water
from any point of the cycle.

For a quantitative assessment of the water cycle on the Earth it is necessary to make
a numerical estimate of the amount of water circulating through it. This can be done
either at the place where water reaches the surface of the Earth from the atmosphere
(precipitation), or where water leaves the Earth's surface in form of water vapour
(evaporation). In both cases the numerical basis necessary for an estimate must be
obtained from observations. On the continents the amount of water vapour precipi-
tated from the atmosphere can be determined with sufficient accuracy by direct mea-
surement of the precipitation, and this quantity can be determined more accurately
the denser the network of rainfall measuring stations. The determination of the mean
precipitation amount over the sea is, on the other hand, very difficult and is never
precise because of uncertainties in the measurement of precipitation on boardship.
On the other hand, the accurate determination of the amount of mean evapora-
tion on the continents is accompanied by considerable difficulty while the direct
determination of the evaporation from the oceans seems possible and can be made
more easily because of the more uniform conditions at this surface, however, in
practice critical examination is still needed. These circumstances give particular
importance to the question of the magnitude of evaporation amount from different
regions of the oceans.

219

220 Evaporation from the Surface of the Sea and the Water Budget of the Earth

2. Direct Measurement of the Evaporation on Board Ship and Methods for Obtaining
Corresponding Values for the Sea Surface

The evaporation from the surface of the ocean can only be measured from ships
under way and this involves — probably even more than in direct measurements ashore
— a number of sources of error that require special attention. Evaporation differs
from other meteorological factors such as barometric pressure, temperature, wind,
and cloudiness, in that the apparatus used to measure it is able to exert a very con-
siderable influence on the observed values. This greatly increases the difficulty of
getting reliable and useful results, because the values obtained are always only relative
vahies which give correct absolute values only after the application of suitable correc-
tions.

Measurements on board a moving ship are made by using a vessel filled with sea-
water and hung in a cardan suspension. Mohn (1883) used a volumetric method of
determining the amount of water evaporated at a given moment. The loss in weight
due to evaporation of the cylindrical evaporation vessel was replaced by refiUing it
with fresh water to bring it back to a zero mark ; the evaporation height could thus be
determined. A more accurate and reliable method is the determination of evaporation
by observing the change in salinity which occurs in the evaporating water as a conse-
quence of evaporation. Following a suggestion of Penck, a cylindrical glass vessel is
used which has a cross-section of 288 cm^ and a volume of 2400 cm^; it is filled with
sea water and placed within a white-painted or nickel-plated mantle that protects the
vessel against direct radiation. Chlorine titration before and after the evaporation
period gives the increase in salinity and allows a very accurate determination of the
evaporation. The mean error in a single measurement is seldom more than 3%; it
derives from the uncertainties of the salinity determination and in refilling the vessel,
while the diminution in volume during the observation can be disregarded.

Denoting with g^ and gg the weights of the sea-water at the beginning and at the end
of the evaporation time, withg^ and^j,, those of evaporated water and of pure water and
finally with gs that of the salt at the beginning of the evaporation, then

gi = gw + gs and gz = (gu, — ge) + gs-
The salinity at the beginning and at the end of the measurement is then

jj = 103 X — ^ — and s^ = 10^ x

gw i g s \gw ge) ~r g s

If p is the specific weight (density) of sea-water at the beginning of the measurement
and J is the volume of the evaporated water amount from the vessel, it follows that,
from the above relations,

Si S<2, ^1

g2 = gi- and ge= pJ — ^ — .

If p is the specific weight (density) of distilled water at the mean temperature t^
at the time of the measurement and o is the area of the evaporating surface, then the
evaporation height becomes

J p S2 — Sj^

«<. = --

op s^

Evaporation from the Surface of the Sea and the Water Budget of the Earth 221

If 30%o < s < 40%o and — 2°C < t^ < 30°C, then with sufficient accuracy
pIP = 1 -027 and with the above values for o and /

So — s-i
h, - 88-3 .

For mean conditions the accuracy of //^ is 3-4%, which is quite sufficient. If systematic
errors in the measurement are avoided (such as spray from the sea, water drips, and
inflow of ash, etc.), the difficulty at the present day is not in getting comparable mea-
surements of evaporation but in the correction of the values in order to obtain sea
surface values. The value required is not the evaporation height in a vessel on board
but the considerably different values at the surface of the sea. For this it is necessary
to know: (1) the factors on which in the most general case the magnitude of the
evaporation depends; and also (2) the differences between these factors in the vessel
on board the ship and at the free sea surface. For an answer to these questions it is
thus essential, during long-term series of observations on board ship, to perform
special additional measurements; for instance, of the temperature of the surface of
the water in the evaporation vessel, etc., in addition to the self-evident meteorological
observations on board. Measurements of the evaporation in this way have not been
made very often. They were first carried out by WiJST (1920, see also Schmidt, 1921)
in a fundamental investigation, and these values after critical interpretation were used
to derive more correct average zonal values of the evaporation at the surface of the
ocean. From all the formulae which have been used many times to calculate the effect
of the meteorological factors, the best is the expanded Dalton evaporation formula
in the form :

he = cf{u){\ + at){0-9Se, - e„).

e^ is the height of water evaporated in 12 or 24 h, c is a constant and f{u) takes into
account the eff'ect of wind velocity. The last two expressions in brackets, which were
termed the "evaporation potential p" by Marvin (1909), take into account the effect
of the air temperature / and of the difference between the saturation pressure of water
vapour at the temperature of the evaporating water eg and the water- vapour pressure
in the air e^. The factor 0-98 takes into account the effect of salinity which hinders
evaporation. As it is known if 30%o < s < 50%o at the sea surface then e^ can
be put equal to 0-98^5 in the atmosphere whereby the evaporation is almost indepen-
dent of the salinity.

From reliable measurements on board a moving ship single values of the quotient
ejp can be computed and can be related with the motion of the air at the time of
measurement, which is identical to the actual wind measured on board the moving
ship. A conversion of the evaporation measured on a moving ship into that which
was measured at deck height with an evaporation vessel at rest and at the true wind
speed over the sea can be done with sufficient accuracy.

For correction of the true evaporation obtained from the instrument on board ship
to that at the surface of the sea, i.e. of the free ocean, Wiist used the gradient of the
meteorological factors between the evaporation vessel and the sea surface. A basis for
estimating the gradients of air temperature, humidity and wind speed immediately
above the sea surface was obtained from observations in the Baltic Sea (September

222 Evaporation from the Surface of the Sea and the Water Budget of the Earth

1919). The mean values of the gradients used by Wiist are, however, obtained from
relatively few observations but appear, as confirmed by later observations, to be of
the correct order of magnitude (Shouleikin, 1928, Montgomery, 1936-7/8, Bruch
1940).
The total reduction factor for a conversion of this kind amounts to

k = 0-48 ± 008.

It is seen that the actual evaporation at the surface of the sea is on the average somewhat
less than half of the true evaporation measured by an evaporation vessel on board ship.
In this way Wiist obtained for the North Atlantic, for example, the following mean
evaporation heights (Table 84) for average meteorological conditions.

Table 84. Evaporation in the Atlantic

Mean evaporation at

Mean

Mean vessel

the sea

surface

wind

evaporation

Climatic regions

Latitude

speed

according to Wiist

According

(km/h)

to Wust

to Liitgens

mm/day

cm/year

Variable winds

50°^0°N.

146

Subtropical region

40°-30° N.

5-8

212

160

North-east Trade

30°-8° N.

7-8

285

128

240

Doldrums

8°-3°N.

5-5

201

115

South-east Trades

3"N.-20=S.

7-3

267

120

225

Subtropical region

20°-40° S.

5-8

212

175

Variable winds

40'^-55°S.

2-8

102

100

This table also gives some idea of the values measured by an evaporation vessel in
different climatic zones and of the meridional distribution of the evaporation amounts
over the Atlantic Ocean. The last column on the right gives values obtained by
LuTGENS (1911) from his excellent measurements of evaporation; due to unsuitable
correction, however, the latitudinal differences are overestimated, especially the evapor-
ation amount in the trade regions, relative to that in the doldrums. The total procedure
of a direct redaction of the observed evaporation on board a moving ship suggested
by Wiist was later again controlled by Cherubim (1931), and he found, after applica-
tion of some refined but not very important corrections, a reduction factor of 0-54
which, however, he multiplied by 1 -08 in order to account for the influence of the
motion of the sea giving the final value 0-583. This latter increase in the size of the
correction factor by about 8% for the motion of the sea, for which there was no ob-
servational evidence, was regarded by WiJST (1936) as unsuitable since there were other
factors, some acting in an opposite direction which had not been taken into account
and of which the magnitude was equally unknown. The uncertainties of the direct
correction are certainly rather large but if the value obtained by Cherubim is taken
as a maximum and that obtained by Wiist as a minimum then a mean of 053 can be
taken at the present time as the most probable correction factor.

Evaporation from the Surface of the Sea and the Water Budget of the Earth 223

3. Meridional Distribution of Evaporation over the Whole Ocean and its Determination
from Energy Considerations

The mean values of the true evaporation for different parts of the ocean which can
be regarded as the direct result of observations have been used by Wiist
to give values for latitude zones of 10° width in the Atlantic and for the total
ocean. These depend on interpolation and in part on extrapolation and can thus be
considered only as a first approximation. The values recalculated with a correction
factor k = 0-53 are given in Table 85. The zonal variations in evaporation, with
pronounced maxima in the trade wind regions and a low value in the doldrums, are
less pronounced in the figures for the total ocean than in those for the Atlantic alone.
Due to the relatively large proportion of the Polar Sea with a low evaporation the mean
value for the Atlantic is less than that for the total ocean. The mean evaporation for
the total ocean found in this way is 93 cm/year or 2-54 mm/day. The limits of error for
this mean value and for the zonal values are about ±12%.

Table 85. Zonal distribution of evaporation

in the Atlantic and for the total ocean
(According to Wiist. (Correction factor k = 0-53.))

Total ocean

Zone

Atlantic

(mean over all
oceans)

(cm/year)

80°-70° N.

70°-60^ N.

60°-50° N.

50°-40° N.

40°-30° N.

107

30°-20° N.

138

130

20^-10° N.

146

133

10°-O° N.

107

112

0°-10°S.

141

125

10°-20°S.

138

133

20°-30° S.

125

30°-40° S.

40°-50° S.

50^-60' S.

60^-70° S.

Mean

The mean evaporation of the total ocean can also be determined by another method,
suggested by Schmidt (1915). It has already been shown in Chapter III/ 1 (see p. 88),
in discussing the heat budget, that evaporation is one of the most important items
(loss) in the heat budget of the sea. From a comparison of the amounts of heat in-
volved in the heat budget for the world ocean the maximum heat amount available
for evaporation can be estimated.

Denoting the mean annual energy gain of the total ocean surface due to sun and
sky radiation by Qs, the energy loss due to outgoing radiation from the ocean to the

224 Evaporation from the Surface of the Sea and the Water Budget of the Earth

atmosphere with Qi,, the loss by evaporation with Q^, and the loss by convection
(turbulent heat conduction) with Q^, then for a stationary state Qs ^ Qb-\- Qe+ Qh-
Introducing R = QJQe and E = QJL, where L is 585 cal/g, the latent heat of
evaporation of water, into the basic equation for the heat budget of the ocean (see
p. 89) then

L(l + R) '

If the radiation terms Qs — Qb and R are known it is possible to calculate the evapora-
tion. Schmidt carried out this calculation using, however, R' == QeliQs — Qb) instead
of R, and determined R' from general considerations as about 0-70. This gives a mean
correction factor k for evaporation measurements on board ship, and he found k =
0-43 as the most probable value. For the extreme case ^^ = 0 (disregarding all con-
vectional processes) Kleinschmidt (1921) found an upper value for k of 0-61. The
good agreement with the value of Wiist of 0-48 is remarkable. Angstrom (1920)
showed that Schmidt's estimate gave too large a value for R. From measurements and
energy considerations he concluded that the value of R is only 0-1, which means that
of the total gain in energy Q^ — Q^, only 10% will be given off to the atmosphere by
convection and approximately 90% used for evaporation.

The method of Schmidt has been carried further by Mosby (1936), who attempted
in particular to remove the uncertainty in the determination of the incoming radiation
Qs by the use of an empirical formula (see p. 91). The values for Q^, thus obtained,
are given in Table 86.

Table 86. Heat budget for the ocean.

(According to Mosby (g cal cm"^ min~^))

Areas of the

Latitude

Qs-Q,

zones in
million km^

70°-60° N.

0040

5-3

60°-50° N.

110

50^-40° N.

150

40 "-SO"^ N.

125

20-8

30°-20" N.

150

25-1

20°-10°N.

167

31-5

10°-O^N.

171

340

0°-10°S.

175

33-6

10°-20° S.

171

33-3

20°-30° S.

150

30-9

30'-40° S.

129

32-2

40°-50° S.

097

30-5

50°-60" S.

067

25-4

60-70° S.

0041

171

The mean value between 70° N. and 70° S., taking into consideration the ocean area
of the separate zones, is estimated to 0-132 g cal cm~2 min~^ Since on average for the
entire ocean advective processes are assumed to be of no importance, this is the
average amount of heat available for evaporation and convection. However, Mosby

Evaporation from the Surface of the Sea and the Water Budget of the Earth 225

could give only an estimated value for the heat contribution to be ascribed to convective
processes, v^hich was based principally on Angstrom's investigation. This quantity
was finally assumed to be about one-tenth of the heat available for evaporation, so that
a heat amount of about 0-119 g cal cm~^ min"^ must be available for evaporation.
The estimation of the convectional flow discussed on p. 92 led to a value of
about 20gcalcm-2day"^, i.e., about 0014 g cal cm-^min-^ The agreement
with the value assumed by Mosby is rather good, but this estimate applies
only to temperate latitudes and the value should be increased for warmer
climates to 0-030 g cal cm ~-min~^ Choosing a mean value of about 0-022
g cal cm"2 min-^ then the amount of heat available for evaporation will be 0-1 II
g cal cm~2 min~^. Since the evaporation of 1 cm^ of water requires approximately
590 g/cal this latter value gives a mean evaporation of 97 cm a year, while Mosby's
value is 106 cm a year. The accuracy here is also scarcely more than 10%. These
values are in good agreement and within the limits of uncertainty of the value derived
by Wiist.

Another possibility for determining the value of R was pointed out by Bowtn
(1926). For identical eddy coefficients for the diffusion of water vapour and the turbu-
lent conductivity of heat, the upward flux of the latent energy of water vapour and
heat are given by

0-621 de ^ d§

Q, = -L -y- A -j-_ and Qn = -c^ A ^-

(see p. 92 concerning the latter equation).
From these equations it follows that

O, 0-62 IL dejdz '

Putting p = 1000 mb and L = 585 and replacing the differentials by corresponding
finite differences the Bowen ratio is obtained:

R = 0-64 -^

es - ea

where t?, and '&a denote the temperatures of water and air and e^ is the maximum
vapour pressure of water at temperature 'Og and e„ is the actual vapour pressure in the
air. Jacobs (1942, 1943) has determined the dependence of the Bowen ratio on latitude
in the North Atlantic and the North Pacific and found that R decreases with latitude.
The following values were found as the mean for both oceans:

Latitude (" N.)

70-60

60-50

50-40

40-30

30-20 20-10

1(M)

0-45 0.31 0-21

015

Oil 010 0 10

The northward increase is an effect of the continents from which the cold air flows out
over the warm sea in the winter. In the Southern Hemisphere this effect is missing
so that R may increase only to about 0-25 at 70° S.

By making proper use of all observations and methods which were more or less
independent on each other, WiJST (1954) has evaluated a mean meridional distribution

226 Evaporation from the Surface of the Sea and the Water Budget of the Earth

of evaporation. These mean annual evaporation amounts together with mean annual
values of precipitations are contained in Table 87.

Table 87. Mean values of precipitation, evaporation and the difference between them

(E — P)for the entire ocean {including adjacent seas)

(According to Wust, 1954)

Evaporation-

Zone in

Precipitation

Evaporation

Precipitation

degrees

cm/year

cm /year

cm/year

70-65 N.

-22

65-60 N.

-45

60-55 N.

-43

55-50 N.

105

-50:

50-45 N.

112t

-46

45^0 N.

102

-18

40-35 N.

108

35-30 N.

125

30-25 N.

132

25-20 N.

57:

137t

80t

20-15 N.

135

15-10 N.

103

132

10-5 N.

187t

126

-6I:

5-0

146

113+

-33

70-0 N.§

1010

110-6

9-6

0-5 S.

105t

125

5-10 S.

109t

137

10-15 S.

139t

15-20 S.

137

20-25 S.

133

65t

25-30 S.

65:

123

30-35 S.

110

35^0 S.

40-45 S.

110

-32

45-50 S.

117t

-61

50-55 S.

109

-70

55-60 S.

12:

-72:

0-60 S.§

91-45

102-1

10-7

t Maxima; : Minima; § Excluding polar zones

4. Geophysical Aspects of Evaporation Problem

Evaporation is a physical process that takes place at the boundary surface between
water and the air above it and depends on the conditions both in the water and in the
air in the immediate vicinity of the surface. The formula showing the dependence of
the evaporation height occurring in a certain time on the meteorological factors is
usually given in the form

hn =f(p) X fiT) X f,(u) X (e, - Ca),

where each term represents the effect of one of the meteorological elements (p the
pressure, T the absolute temperature, u the wind speed); e, is the maximum vapour

Evaporation from the Surface of the Sea and the Water Budget of the Earth 227

pressure corresponding to temperature and salinity of water, Ca is the vapour pressure
in the air. Different expressions have been chosen for the functions /i, /g and/3 ^nd a
formula of this type is given on p. 220 which shows the dependence of observed
evaporation on the prevailing meteorological conditions and with a suitable choice of
constants gives satisfactory values. However, it can hardly be assumed that such an
evaporation formula which is a product of different functions could give a correct and
causative description of the actual physical process of evaporation ; it is rather to be
expected that such a formula would be of the form

fh = fiP, T, u) {e, — e^,

where the function/is probably a complicated function of the meteorological factors.
According to the results of research in turbulence, the transport of the water vapour
continuously formed at the sea surface into the air immediately above it proceeds by
turbulent exchange; the magnitude of this exchange depends on the roughness of the
evaporating surface which in turn also depends on the velocity of the air over the water.
SvERDRUP (1936, 1937-8, 1951) was the first to attempt to clarify the problem as to
how the evaporation process operates at the surface of the sea with a well-defined
roughness under the influence of the turbulent exchange. His ideas are based on two
circumstances which aie essential for a solution of this problem:

(1) Immediately above the water surface a thin boundary layer exists in which the
water vapour transport proceeds only by ordinary (molecular) diffusion.

(2) Above this boundary layer the water vapour transport proceeds through the
turbulent exchange A in form of random movements of the air particles (turbulence).

The exchange A (according to laboratory experiments) is a linear function of the
height above the water surface and depends on the roughness of the water surface.
The latter is described by the roughness parameter Zq, and according to the results of
Rossby about the increase of wind velocity with height Zq is considered constant im-
mediately above the sea surface (zq = 0-6 cm). This is valid for weak to moderately
strong winds. Correspondingly,

A = pk^iz — Zo) J- ,

where r is the tangential force (stress) of the wind, p is the density of the air and kg
is the Karman constant with a value of 0-38-0-40 (see Vol. I, Pt. 2).

The thickness of the boundary layer immediately above the water surface depends on
the wind velocity. The layer itself can hardly be regarded as invariably composed of
the same air particles. Since the turbulent eddies will sometimes penetrate down to and
into the boundary layer, it must clearly be understood that this layer occasionally
disappears completely; however, after some time it will always be re-formed so that
a mean thickness of this layer can be introduced.

In addition to the theoretical case built up on the basis of these ideas Sverdrup
also discussed a second possibility where the water surface was assumed to be "smooth"
and the transport of water vapour away from the sea, due to turbulence, starts from the
sea surface itself. Observations seem to favour the first case with a diffusion layer and
turbulent transport above, and therefore only this case will now be dealt with.

For the exchange coefficient A we may write

A = pko(z — Zo) u^.

228 Evaporation from the Surface of the Sea and the Water Budget of the Earth
if the so-called 'friction velocity'' is introduced according to Karman:

The values for Zq and u^ follow from measurements of the wind over the surface and
depend on the character of this surface.

In the turbulent layer the water vapour transport E (expressed in g cra"^ sec~^)
directed upwards is due to the turbulent exchange process and is given by

where q is the specific humidity which decreases upwards, q may be replaced in this
formula by the vapour pressure e according to the well-known formula

0-623

and one obtains with sufficient accuracy

0-623 de

E = A -T .

p dz

The process of evaporation must be regarded as stationary (E = constant), so that
with the above value for ^, if c is a constant,

de c

dz (z + Zo) *

Denoting the value of e at the lower boundary of the turbulent layer or at the upper
limit of the diffusion layer (thickness d, z = d) with e^, then integration gives

1 - + -0

''-'^-'^''d^rj-^'

On the other hand, the quantity E was found to be

0-623 . , . ^ dz 0-623

E= -— pkou^iz -}- Zo)~ = --^7- pkou^c.

The transport of water vapour through the diffusion layer is given by the equation

es — ea

where S is the diffusion coefficient of water vapour in the boundary layer with
reference to vapour pressure. At the boundary of the two layers the water vapour
transport is steady so that the necessary condition

z = d, E = E'

must be satisfied. Considering, in addition,

0-623

b = K p.

Evaporation from the Surface of the Sea and the Water Budget of the Earth 229

where k is the diffusion coefficient in cm^/sec then the thickness of the diffusion layer
is given by

'e.

and for a roughness parameter Zq

d +

•]

A-o

0-165

u^. =

Uz,

hl{(z + Zo)/Zo} "^ log{(z + zo)/zo}

where «, is the wind velocity at a height z. Finally, the evaporation E is thus obtained
from the above formula

E =

8u,

(es - e^).

If the thickness of the diffusion layer is known then the evaporation E can be calcu-
lated, if we observe: (1) the wind velocity at a height above the surface of the water,
by means of which u^ is found; (2) the temperature and the relative humidity at this
height, wherewith e^ is known; (3) the salinity, from which e, can be determined. Only
observations can give information on the thickness of the layer d. For this Sverdrup
used the values determined by Montgomery (1940) on board the research vessel
"Atlantis", wheieby Zq = 0-6 cm was assumed. Table 88 contains this calculation.

Table 88. Values of the friction velocity u^, the evaporation E and

the thickness of the diffusion layer d for a rough water surface

(zq = 0-6 cm)

(According to observations of the research vessel "Atlantis")

Observation

«*

10«£

group

(cm/sec)

(g cm"- sec"^)

^u* ■" ^«cm

(cm)

13-2

106

0-30

0-28

«i

14-3

1-34

0-34

0-22

170

1-98

0-30

0-33

18-7

2-86

0-33

0-31

24-8

5-15

0-39

0-29

25-3

4-64

0-45

0-23

25-6

5-82

0-53

016

29-2

8-49

0-74

009

29-2

5-68

0-70

010

36-3

6-98

0-80

010

The value of d decreases with increasing wind velocity, and Fig. 103a shows that as a
rough approximation d increases linearly with 1/w^. With suitable weighting of each
group Sverdrup obtained d = 4-12/z/^.

Unfortunately, there are no simultaneous measurements of evaporation available
to allow a close test of the theory. Sverdrup with these values of <y and using the meri-
dional distribution of temperature, relative humidity and wind velocity at the surface
of the Atlantic, calculated the meridional distribution of evaporation and compared
this theoretical distribution with the zonal values obtained by Wiist, applying the

230 Evaporation from the Surface of the Sea and the Water Budget of the Earth

cm
0-30

0i20

0-0

Yvr 0-01 0-02 0-03 004 0-05 0-06 0-07 0-08

Fig. 103a. Relationship between the thickness of the diffusion layer (d) and the reciprocal
of the shearing-stress velocity (1/w*).

correction method to the direct measurements of the evaporation on board ships.
Taking, according to the preceding section, the mean annual evaporation height
for the Atlantic as 100 cm then the values given by Wust,and shown in Table 89 having
a mean evaporation of 83 cm, must be multiplied by the factor 1-22 (they have to be
divided by 293 if values in mm/day are needed).

Table 89. Vahies of evaporation for zonal regions of the Atlantic found by calculation
from the meteorological data and from observations of evaporation

Evaporation

Latitudinal

Temperature (°C)

Relative

(mm/day)

humidity

^.t

zone

Calcu-

Ob-

Water

Air

(%o)

(mb)

(m/sec)

lated

served

50°^0° N.

10-8

10-5

12-74

1010

8-4

2-3

2-2

40°-30° N.

18-3

17-2

20-64

15-73

6-7

3-3

3-2

30°-8°N.

25-4

24-9

31-82

24-92

6-7

5-2

4-3

8°-3°N.

27-4

26-8

35-80

29-26

2-8

1-9

3° N.-20" S.

25-8

25-7

32-62

25-80

4-1

4-0

20°^0" S.

19-5

18-3

22-25

16-85

5-6

3-2

40°-55° S.

9-9

8-7

12-00

9-26

7-8

2-2

1-6

t Taking into account the factor 0-98 as the effect of salinity

Table 89 presents this calculation, and a comparison between calculated and
observed £■ values. Figure 104 shows the results graphically. The agreement between tne
observed and the calculated values is rather good, and in any case considerably better
than in the second case treated by Sverdrup for a smooth surface without any diffusion
layer. This agreement shows that the theory of a diffusion-layer with a thickness de-
creasing with increasing wind velocity and with a turbulent layer above it with a
roughness parameter Zq = 0-6 cm is capable of explaining the evaporation at the sur-
face of the sea. However, all the assumptions so far are based on very few observa-
tions, so that further support by systematic measurements would be extremely de-
sirable. The evaporation formula on p. 228 shows in any case that the dependence of
the evaporation on the meteorological conditions in the atmosphere above it is more
complicated than was assumed in previous relationships, and that deeper insight into
these phenomena can only be obtained by a geophysical analysis of the evaporation
process.

Evaporation from the Surface of the Sea and the Water Budget of the Earth 231

50° N 30° 20° 10° 0° ICP 20° 30° S 50°

Fig. 104. Calculated and observed values of evaporation in successive latitudinal zones of
the Atlantic (according to Wiist).

The formula given above for E can be changed into a more practical form. Ex-
pressing e in mm and u in m/sec and putting p = 1000 mb and z = 10 m then E
in mm/24 h vi'ill be given by

E = K^oi^s - ^lo) "lo-

Kio lies between 0-12 and 0-19. This simple formula is quite remarkable since it is
based solely on theoretical considerations. The theory of evaporation discussed above
involves a hydrodynamically smooth surface with a laminar boundary layer (molecular
diffusion of water vapour) with a turbulent layer of air above it. The evaporation
from the water surface can also be calculated for other different stratifications of the
layer of air above the water and there can be obtained the general equation

E = pKoyF^^es — e^) Ua,

where kq is the Karman constant, y is the frictional coefRcient and F is the Mont-
gomery evaporation factor. The latter depends on the structure of the lowermost
layer of air, on the wind velocity and on the stability and nature of the boundary
layer. The observations of Montgomery seem to indicate a sharp increase in F at
u = 6-5 m/sec for a = 6 m, while lower values for F are found for lower wind veloci-
ties. In this case, the water surface is smooth and has a laminar boundary layer above
it. Turbulence only becomes effective with higher wind velocities and increases the
evaporation rate. Only further observation can show whether this transition is
gradual or sudden. On this subject see Vol. I, Pt. 2, and especially Munk (1947).

5. The Water Budget of the Earth

The source of atmospheric water vapour is to be found in the first place in the
evaporation of water at the surface of the ocean ; the evaporation of water from the
continents taking only a secondary place. In so far as the water vapour formed over
the surface of the ocean remains there, condenses to clouds and returns as precipita-
tion, it is referred to as a minor water cycle. Part of the oceanic water vapour is,
however, carried by air currents inland over the continents and together with water
vapour originating from the land gives rise to precipitation over the land. If this water
is not evaporated again and returned directly to the atmosphere, it will be returned
to the sea by streams, rivers and ground water (run off), and closes in that way the
major water cycle.

232 Evaporation from the Surface of the Sea and the Water Budget of the Earth

Since the total amount of water on the Earth from a more general view-point can
be considered a constant, seven items enter into the total water budget, which for a
stationary state must be related with each other according to a strict principle of
dependence.

These seven items are the following:

Eq the mean annual evaporation amount from the oceans ;

Ec the mean annual evaporation amount from the continents;

Pq the mean annual precipitation amount over the oceans;

Pc the mean annual precipitation over the continents;

Wq the annual amount of water vapour in the atmosphere above the sea passing
over to the continents ;

Wc the annual amount of water vapour in the atmosphere above the land passing
over to the sea ;

R The annual outflow of water in rivers, etc., into the sea (run off).
The constancy of total water in all oceans requires that the total mean annual inflow
of water into the oceans Pq -\- R must be balanced by the total amount of water re-
moved £"0; the constancy of the water on the land requires that the water gained by the
land Pc must be equal to the water lost E^ + R; and finally the constancy of the at-
mospheric water vapour over the oceans and over the continents requires that

E^-W,+ W, = Po,

and

E,+ W,-W,^ P,.

From this it follows that the annual outflow of river water and other water into the
ocean (total run off) must be exactly equal to the difference between the amount of
water vapour in the atmosphere passing from the sea into the land and that passing
from the land out over the sea. Thus, the following formulation for the balance of
the budget of the water cycle on the Earth is obtained, which are known as the basic
"Bruckner" equations for the water balance of the Earth (Bruckner, 1905; Fischer.
1925).

These basic equations can also be derived, as has been shown by Defant and
Ertel (1943), from the continuity considerations of the total water content of the
atmosphere in a closed and more general form. The amount of water contained in a
unit volume of atmospheric air consists, on the one hand, of the dry air (density:
Pa g/cm^) and of the amount of water vapour (density: p^), and on the other hand, of
the amount k (g/cm^) of the condensed water vapour in liquid or solid form. Changes
in the amount (p^) of water vapour in unit volume and unit time can occur locally:

(1) By the convergence (negative divergence) — div {pw'm) of the convectional
flow p^tt) of water vapour.

(2) By the convergence of the turbulent flux — div S. The turbulent flux is given
by (5 = —A grad q, where A is the exchange coefficient (g cm~^ sec~^) of the
specific humidity q = {0-623lp)e.

(3) By evaporation of a definite amount of condensate or by condensation of a
definite amount of water vapour, respectively:

±m [g cm~^sec"^].

Evaporation from the Surface of the Sea and the Water Budget of the Earth 233

This quantity m represents the internal turnover of water in unit volume per unit time
and is positive if more condensate evaporates than water vapour condenses, but
negative if more water vapour condenses than condensate evaporates.
The continuity equation for water vapour is thus

-^ + div (pu>^) + div S = +m.
ot

Local changes in the condensate k in a unit volume in unit time can occur in two ways :

(1) By the evaporation of a definite amount of condensate or by condensation of a
definite amount of water vapour respectively. If more condenses than evaporates,
then according to the above argument this change is -\-m; however, in the opposite
case, —m.

(2) The water content in a unit volume (liquid or solid) can also change if, for
instance, part of it is removed as precipitation or is advected by air currents to other
levels. For each point in space this movement of condensate can be considered a
condensate flow which can be described by a vector 51, The absolute value |Sl| is the
amount of condensate which passes in unit time through a unit area of a surface
perpendicular to the direction of movement of the condensate. At the point where
there is no condensate or if the condensate shows no movement then 1^| =0. The
flux of condensate through a unit surface along the normal /m is 5l„ = 51^, and in par-
ticular, for z = 0

gives the precipitation amount per unit area and unit time at the surface of the Earth
(z = 0). The change of k due to such processes of condensate movement is then given
simply by the convergence —div £ of the condensate flux.

The condensate continuity equation is then

— = —div ^ — m.

Adding the two continuity equations for water vapour and condensate gives the
continuity equation for the total water content finally in the form

^'''""' + "^ + div (p„tt, + S + S) = 0.

For a stationary, average state in the atmosphere this equation reduces to

div (p^lt) + S + SI) = 0.

Imagine now a vertical surface of control B, which parallels the coasts of a (not neces-
sarily continuous) continent and reaches upwards to the upper limit of the atmos-
phere. Considering a surface element dB with a horizontal normal n directed towards
the interior of the continent (landwards). Then, integrating the above equation over
the total volume between the surface of control B, the surface of the Earth and the

234 Evaporation from the Surface of the Sea and the Water Budget of the Earth

upper limit of the atmosphere, it follows, according to the Gaussian integral law,*
that for the total column of air over the land

(p^rt) + Q)dB -\\ (B,dL- I ^,dL = 0,

where dL is an element of the land surface L ; the two terms of the first integral vanish
for the land surface and the upper limit of the atmosphere since at these extreme
limiting surfaces either p„ or it) will be equal to 0 or .4 will be 0; for the two other
integrals the amounts passing through the control surface B disappear. Now, the mean
precipitation amount per unit time on the continent is

^,dL

P dL,

and the mean evaporation amount over the continent is

Finally, the water vapour flux through the surface B towards the land is

S,dL.

W.- Wr

(Pu,^ + S)„ dB.

The condition of a stationary state thus gives one of the basic Bruckner equations as

Ec~Pc + {W^ - W,) - 0.

If, on the other hand, the integration is taken over the total ocean, one obtains
in the same way [the inwards (oceanwards) directed horizontal normal of dB is now
—n] the second basic Bruckner equation

P, + E, + {W^ - W,) = 0.

Integration of the continuity equation over the entire atmosphere above the surface
(C + O) gives

Pe i Pq ^^ Ee Eg,

which can, of course, also be obtained by subtraction of the first two equations.

The basic equations for the water budget of the Earth involve five quantities; a
knowledge of three is sufficient to evaluate the others numerically. In general, it does
not matter which of them we presume as known and which we want to obtain. How-
ever, the accuracy with which the different quantities can be determined from the
available observations is not the same for each. The precipitation over the sea can be
estimated only with difficulty. For that purpose in wide regions of the oceans only the
rain density (the mean precipitation amount for a single rain day) and the rain fre-
quency (the average number of days with precipitation) are available from ships'

* The Gaussian integral law states that the volume integral of a volume Kwith a surface A taken
over div a is equal to the negative surface integral of r„ taken over the entire surface A, where //
is the normal to A directed towards the interior, so that

III 1.va</^'=-||».,/^.

Evaporation from the Surface of the Sea and the Water Budget of the Earth 235

observations, and the product of these quantities gives only an approximate idea of
the annual precipitation amount. In addition to these observations there are also
available precipitation records in coastal areas or from islands, but these are often
strongly affected by local topography so that there is usually a greater precipitation
amount than over the neighbouring oceanic regions. Therefore, the utmost critical
inspection and caution is needed in the use of records of coastal and island precipita-
tion for the construction of isohyeths for the oceans. However, with suitable allow-
ances, better numerical estimates can be made of the evaporation over the sea so that
the mean evaporation amount allows in return an estimate of the precipitation amount
over the oceans.

The reverse applies on the land ; here the determination of the mean evaporation
involves almost insuperable difficulties, but a dense network of precipitation stations
can give the mean precipitation with suitable accuracy. In this way a complete picture
of the water budget of the sea, the land and of the total Earth can be obtained. Such a
summary however, does not give absolutely correct annual values since the accuracy
of each item in the water budget is not very great. It is thus of more importance to
enclose the different values within the most narrow limits possible so that the indi-
vidual values either support or exclude each other, in order to obtain maximum
probability. Table 90 summarizes the essential characteristics of the water budget
of the Earth.

Table 90. Most probable water budget of the Earth

Precipitation

Evaporation

Outflow (-) and
inflow (+)

10' km' /year

cm /year

1 0' km'/year

cm /year

10' km'/year

cm/year

Ocean
Continent

Entire Earth

324
99

423

90
67

361
62

423

100

+37
-37

+ 10
-25

The following points may be noted. The figures for precipitation are based on those
obtained by Meinardus (1934) from a most detailed investigation of the distribution
of precipitation over the Earth based on the precipitation charts for the oceans pre-
pared by Schott. The values for the land were taken directly. For the sea a correction
was applied based on the criticisms made by Wtisx (1936) of these charts. Meinardus
found a total precipitation over the oceans of 411-6 x 10^ km^ which corresponds to
a mean annual rainfall of 114 cm/year. The mean precipitation over the oceans
would thus be 1-7 times greater than over the land (67 cm/year), which can hardly
correspond to actual conditions. No doubt too much consideration of island and
coastal precipitation must have appreciably raised the precipitation amount over the
oceans. Calculation of the precipitation over the sea from the total evaporation over
the sea of 100 cm/year = 361 x 10^ km/year and the inflow from the land (Fritsche,
1906) of 37-1 X 10^ km^/year gives a correction factor of 0-79 for correcting the rain-
fall at coastal and insular stations to values for the undisturbed sea surface. The
coastal and insular values are thus on the average raised by about 20% by the effects

236 Evaporation from the Surface of the Sea and the Water Budget of the Earth

of topography above the values for the open ocean. This probably is not too far from
the actual conditions.

For the entire Earth, according to recent calculations (Reichel, 1952), the
mean annual precipitation is about 86 cm/year, which under stationary conditions is
balanced by an equally large evaporation amount. Therefore the average evaporation
amount for the whole Earth amounts to 2-37 mm water per day. An interesting graphi-
cal representatation of the total hydrologic cycle has been given by Lettau (1954),
and is shown in Fig. 105. It gives detailed information on all aspects of this cycle.

ATMOSPHERE

L I THOSPHERE

Fig. 105. Schematic diagram of the hydrologic cycle. 100 relative units = 85-7 g cm~* year"
or 857 mm global annual mean of precipitation (according to Lettau, 1954).
1 ^ , Evaporation; 2 • • •■► , Precipitation; 3 _ — — > , Dew deposit;

of water vapour; 6 e = values smaller than 0-5 rel. units.

4 o o o o ot> , Run off;

Removal from and addition to horizontal advection

6. Energy Budget between Ocean and Atmosphere for DiflFerent Oceans and Oceanic
Regions

The heat turnover between the total ocean and the total atmosphere has already been
discussed in previous chapters. It is also of considerable interest to know the energy
budget between the ocean and the atmosphere for the individual oceans and for differ-
ent parts of the ocean, since on this depend the effects of the sea on the atmosphere
above it or, in turn, the influences of the atmosphere on the sea. Such investigations,
in spite of their importance, have only recently been made and indeed have been car-
ried out almost exclusively by Jacobs (1942, 1943, 1951^, h) and Albrecht (1949,
1951). These investigations are based on the calculation of the evaporation from the
formula on p. 230 using the differences ^^ — 'da and e^ — e^ derived from climatologi-
cal charts of the oceans. Doubts about these latter values have been expressed by
Dietrich (1950), but it appears that any errors that may have been introduced in this
way are not systematic but may vary from one region of the sea to another and
should, at least in part, cancel out. Calculations of this type have been made especially
for the North Atlantic and the North Pacific, for which the climatic charts are more
reliable. Such calculations of course give only a rough estimate but they serve, however,
to give an approximately quantitative idea of the interplay between ocean and atmos-
phere. At first the most important is the pure heat gain by the radiation turnover
Qs — Qb, whereby Qg is the absorption of solar and sky radiation and Qi, is the radia-
tion loss from the sea surface.

Figure 106 shows the geographical distribution according to Sverdrup (1943) of the
annual surplus of radiation penetrating the water surface. Over the whole year the
oceans have everywhere a gain of heat from radiation, but north of 25° N. this gain
decreases rapidly with latitude, therefore from 10° to 45° N. it is smaller on the eastern

Evaporation from the Surface of the Sea and the Water Budget of the Earth 237

238 Evaporation from the Surface of the Sea and the Water Budget of the Earth

Evaporation from the Surface of the Sea and the Water Budget of the Earth 239

240 Evaporation from the Surface of the Sea and the Water Budget of the Earth

Evaporation from the Surface of the Sea and the Water Budget of the Earth 241

30^ 40^
N Lot

Fig. 1 10. Energy interchange between the sea surface and the atmosphere at different seasons
of the year (1) between 35° and 40° N. in the Pacific and Atlantic Oceans; left side: North
Pacif.c Ocean, rght side: North At'antic Ocean. (2) along the western (left side) and eastern
(right side) sides of the North Pacific Ocean; (3) along the western (left side) and eastern
(right side) sides of the North Atlantic.

Dec. Jan. Feb.

Mar. Apr. May

Jun. Jul. Aug.

Sep. Oct. Nov.

242 Evaporation from the Surface of the Sea and the Water Budget of the Earth

side of continents than on the western side. This is mainly due to differences in
cloudiness.

The annual heat loss by evaporation, Q^, according to Jacobs (1951) is given in
Fig. 107. Evaporation is particularly large in the v^estern parts of the two oceans,
where the currents carry warm water northward (Gulf Stream and Kuroshio). It is,
however, less in the eastern parts where there are cold currents flowing southward.
The extreme seasons show considerable quantitative differences in evaporation. In
middle and higher latitudes the evaporation is large in winter and small in summer,
but conditions may be rather complicated on the western sides of the oceans where in
winter cold air is advected out from the continents over the warmer sea.

The heat loss Q^ of sensible heat by convection is shown in Fig. 108 for the same
oceans. Also one notices here a distinct increase on the western sides of the oceans
which is of the same type as in the distribution of Q^. In the over-all distribution of
energy given off from the sea as heat, the values for the evaporation predominate and
set the basic pattern. With respect to seasonal changes also the behaviour of both loss
items is rather similar. The sum —Qa = {Qe+ Qn) gives a final value for the total heat
turnover as far as it applies to a current-free ocean. If currents are present then the
equation Qv= Qr— Qa must apply, where Q^ is the energy surplus which is obtained
by each cm^ of the surface under influence of a complete heat exchange with the
atmosphere. This energy surplus, when positive, is carried away from the water mass
unber consideration by currents and mixing processes and represents that part of the
radiational gain Qr which is stored in the water. A negative surplus implies that energy
is supplied to the water mass by currents and mixing processes which then is dissi-
pated by the excess in radiation into the atmosphere (Sverdrup, 1945). Figure 109 shows
the total energy surplus of the oceanic water (g cal cm^^ day"^), and shows that in
the water of larger ocean surfaces, especially in middle and lower latitudes along and
near to the western coasts of continents, some energy is stored in the water while
enormous amounts of energy are dissipated (lost by the ocean) in the Gulf Stream and
Kuroshio systems. Thus, to a very noticeable extent, the areas in which large amounts
of energy are available to the atmosphere are localized in definitive oceanic regions.
Comparison of Figs. 109 and 107 clearly shows that the pattern of total energy ex-
change corresponds to that of evaporation. In order to recognize the seasonal varia-
tions in the energy turnover, it is of advantage to compare these quantities along
definite latitudes or meridians along the eastern and the western sides of the oceans
respectively. This can be seen from Fig. 110. The first diagram shows a marked con-
trast between western and the eastern sides of the oceans. A narrow band representing
the energy loss appears along with the Gulf Stream in the North Atlantic, while a
corresponding and more broad band is connected with the Kurishio. This is under-
standable from the direction of the two currents in the zone between 35° N and 40° N.
The contrast of the two sides of the ocean in a meridional direction is shown in the
other two diagrams. Along the western sides at all times of the year, except in summer,
the largest amounts of energy are given off between 25° N. and 40° N to 50° N. Along
the eastern sides there is a winter minimum in these latitudes. These energy transports
arc undoubtedly of decisive importance for the climatic conditions in the effected
regions and form the basis of the study of the inter-relation between ocean and atmos-
phere.

Chapter VIII

Ice in the Sea

Extensive icefields cover the polar seas. The outer boundaries where the ice borders
upon the warmer surrounding waters of lower latitudes are subjected to a constant
change due to the freezing and melting process. They may take a wide variety of forms
depending on the given external conditions. A plastic and lively description of the
magic of the polar ice world has been given by Weyprecht (1879). Besides the so-
called sea ice, formed by the freezing of sea-water, other floating ice is introduced to
the sea from the neighbouring land by the great rivers (river ice), and in addition
icebergs from the glaciers reach the sea. Floating river ice is comparatively unim-
portant, except in coastal Siberian and North American waters, therefore; sea ice
and icebergs dominate ice conditions in the Arctic and the Antarctic, and may be
carried by ocean currents to warmer oceanic regions. This ice drift prolongs the
existence of the winter ice barrier in the polar regions into spring and summer, and is
thus of considerable importance for navigation.

1. Formation and Terminology' of Sea Ice

Ice crystals are formed in the water either on crystallization nuclei, which are the
smallest possible particles of organic or inorganic origin that are always present, or at
an aggregation of several molecules which meet each other grouped more or less by
chance giving a configuration favourable for crystal formation (Nernst, 1909). It
appears that the triplex molecules are decisively engaged in the first phase of ice
formation. Besides the crystallization nuclei, supercooling of the water is also necessary.
The greater the purity of the water and the less disturbed it is, the more supercooling
is needed. In natural waters there are always sufiicient crystalhzation nuclei present,
and the water is usually in movement so that a very small degree of supercooling of
only some hundredths of a degree Celsius is required to initiate ice formation. How-
ever, supercooling has to be continuous for the formation of ice crystals.

Since the formation of ice releases a latent heat of 80 g cal/g, for a change of water
into ice heat must be continually removed by an amount greater than the latent heat.
The more intensive the cooling and the less disturbed the water, the smaller are the
ice crystals so formed, which show a needle-like structure. If the water is in movement
then the forming ice particles lose this needle-like character, looking then like
flat plates with irregular rounded edges, about 2-A cm long, 0-5-1 cm wide and
0-1-1 mm thick. They usually accumulate and form muddy clumps.

The dependence of the freezing point on the salinity is discussed on p. 45. Only
pure water is involved in the actual freezing process. Part of the salt content of the water
is separated during the formation of the ice and, as a more or less concentrated salt
solution, fills the small separating layers between the ice crystals which themselves

243

244 Ice in the Sea

consist of pure water. As the ice crystals grow they withdraw pure water from this
enclosed salt solution, which thus becomes more concentrated and of more specific
weight; it gradually percolates out between the ice crystals and increases the salinity
of the surrounding water. This diffusion process beneath a forming ice layer is pre-
sumably the reason why the crystal plates in sea-water are always oriented perpen-
dicular to the freezing surface, while in fresh water they are parallel to it. The arrange-
ment of the crystal plates is in similar groups and they are oriented approximately
parallel, relative to each other, so that the structure of simple sea ice is fibrous;
therefore the fracture surfaces of the ice lumps appear perpendicular to the surface of
the ice layer.

The classification and terminology of ice formation and ice forms can be made
according to diff"erent viewpoints ; unfortunately there is still no uniform terminology.
Drygalski (1930) has given a completely general classification of ice forms based on
genetic relationships. The two main forms of ice are shelf ice and sea ice. Shelf ice
represents a transitional stage between the forms of ice occurring on the land and those
found at sea. It Ues along the coast over the continental shelf and is for the most part
a mixture of sea ice and land ice (coastal snow ice). Shelf ice reaches its greatest thick-
ness and extent around the Antarctic; a typical example of this type is that found along
the northern coast of Grant Land which is known as palaeocrystalline ice (Ureis).
Other forms of Arctic shelf ice are found along the east coast of Greenland (Wegener,
1902, "floating land ice"). In sea ice there occurs a gradual change of the ice crystals
to pap ice (ice mud, ice slush); in calm weather and at low temperatures it freezes
together to a hard layer of ice up to 5 cm thick and forms, especially at the surface,
a weakly saline top layer. In a rough sea and at still lower temperatures small sheets
of ice are formed which grow rapidly and assume a plate-shaped form with upwards
bulging edges (pancake ice). The individual plates have a diameter of 0-5-1 m, with a
maximum of about 3 m. In calm weather pancake ice and ice slush freezes together
to form a solid layer of young ice with a thickness of between 5 and 20 cm, having a
greenish blue colour; the surface is wet and still rather plastic. Further growth gives
sheet ice, often forming large lumps which are broken and piled up by pressure forming
pack ice.

A detailed terminology of ice forms has been given by Maurstad (1935; see also
ZuKRiEGEL, 1935). Sea ice is divided according to age into two groups: winter ice
(including young ice) and polar ice. The first is not more than one year old, still rela-
tively soft and plastic, and usually occurs in the form of ice lumps. Polar ice, on the
other hand, is mostly two or more years old, contains little salt and is therefore hard.
Due to ice pressure it soon takes the form of pack ice.

With reference to its position and movement Maurstad distinguishes between solid
ice and drift ice. The first is found for the most part in bays, fiords and above shallow
waters. Also winter ice, as long as undisturbed, may remain stationary during the
entire winter; however, it is usually broken up by long open cracks and drifts away.
Drift ice can take all forms and reaches its greatest extent in the drifting ice fields of
polar ice in the Arctic.

In spring and summer, under influence of the increasing solar and sky radiation
and the warm winds, the winter ice begins to melt. The volume of the salt solution
enclosed in the ice increases and the inner structure of the pure ice crystals is weakened.

Ice in the Sea

245

The ice melts in this way from the interior outwards and becomes "putrid". The sur-
face takes on the appearance of a honeycomb (cells), and the entire mass of ice soaks
through down to a considerable depth. In contrast to the ice formed from pure water,
sea ice has no definite melting temperature, but begins to melt as soon as the tempera-
ture starts to rise. Putrid ice breaks up easily, exposing a much larger surface to the
effects of solar radiation and to warmer sea-water in which it is floating. Most of the
winter ice melts in summer, but a large part still remains, especially along the edge
of the Siberian Shelf, that survives the summer and then becomes polar ice and in
consequence is explosed to a strong annual melting cycle.

2. Physical and Chemical Properties of Sea Ice

{a) The Salinity of Sea Ice

The salinity of sea ice is defined as that quantity of sohd matter (in g) remaining
after evaporation of 1000 g of melted sea ice. The limitation that was found essential
in the definition of the saUnity of sea-water (see p. 36) thus also applies here. The
essential difference between the salinity of sea-water and that of sea ice is that the first
is a rather conservative property of sea-water; while the second, in strict contrast, is
a very rapid changing quantity for each single piece of ice. Nevertheless, the sahnity
of a sample of ice shows only minor variations. This has been shown by the numerous
analyses made by the "Maud" Expedition, 1918-25 (Malmgren, 1927). As has been
noticed by all polar expeditions the surface of young ice is covered by a surface salt
solution, which remains liquid even for low temperatures and keeps the surface of the
ice continuously wet. For very low temperatures only this layer also freezes, giving
a mixture of ice and salt crystals which isolate themselves in form of snow-white
clusters.

Beneath the surface a part of the salt solution remains enclosed between the ice
crystals and determines the salinity of the sea ice. Its amount depends on the processes
going on during the ice formation, specifically on three factors: (1) on the salinity of
sea-water from which the sea ice was formed; (2) on the rapidity of ice formation;
and (3) on the age of the ice. Referring to the first, the salinity of sea ice is less than
that of sec-water, since the part of the salt solution between the ice crystals is always

Table 91

Air temperature ("C)

-16

-28

-30

-40

Salinity of young ice (%o)

5-64 8-01

8-77

1016

Table 92

Ice thickness below
the ice surface (cm)

1
6 13

Salinity (%«)

6-74

5-28

5-31

(3-84)

4-37

3-48

3-17

able to escape. In the analyses of young ice samples made during the "Maud" Expedi-
tion the salinity of sea ice reached a maximum value of 14-59%o, but usually the salinity
of sea ice was between 3 and 8%o. Referring now to the rapidity of ice formation it

246

Ice in the Sea

shows that the faster the ice is formed (at lower temperatures) the less salt solution
can escape and the higher therefore the salinity of sea ice (Tabic: 91). Since ice is
formed more slowly in the deeper layers than at the surface some dependence on depth
can also be expected. For a young ice layer that began to freeze in November 1924,
Malmgren found in April 1925 the values shown in Table 92. Referring finally to the
age of the ice, the older the ice the smaller its salinity. The salt solution leaks through
continuously and this process is accelerated by changes in temperature. Blocks of
ice lifted by the pressure of the ice become almost completely salt-free in the summer by
this process of deconcentration, and can be used after melting for drinking water.

The changes in salinity in winter ice occurring during the course ola year have been
summarized by Malmgren in a diagram given in Fig. 111. The ice formed in October
gradually increases in thickness, and initially the salinity decreases from the surface
downwards. Corresponding to their age the middle layers have the lowest salinity.

0 12345678 9

Fig. 111. Salinity changes in winter ice during the course of the year (schematic, according

to Malmgren).

but at the lower surface of the ice layer the salinity again increase:* since the water
freezes here from below. This is due to melt water sinking below the ice layer and
freezing again immediately due to temperatures below freezing point (about — 1-6°C).
In a dilute aqueous solution freezing proceeds with the formation of pure ice only
until the eutectic point is reached, the concentration of the solution increasing at the
same time. This critical point depends on the salts dissolved in the watoi . When sea-
water freezes the separation of the salts dissolved in the water begins only at — 8-2°C.
For sea-water the situation is simplified only in so far as: (1) all types of water have the
same salt composition; and (2) ice is always the first substance to freeze out. In conse-
quence, no matter how great the salinity, the freezing process always proceeds in the
same way. For a given temperature the concentration and the composition of the salt
solution is to a close approximation the same for all types of sea-water, regardless of
their original salinity (Malmgren). If Tj is the freezing temperature of 1 g of sea-
water of salinity S, then for a temperature t between t^ and --8-2°C only pure ice
will separate out according to the above discussion, and at this temperature there
will be a-r g of pure ice and (1 + a^) g of salt solution. If the sali ity of the salt solu-
tion is S-T, then necessarily

(1 -f a;)S. = S.

Ice in the Sea 247

For sea-water of salinity 5" there will be a similar relationship

(1 + a.')S.' = S'.

Since Sr = St' it follows

S S'

-I = ~ = const.,

\ — a-r I — Or'

that means, the amount of salt solution per gramme is proportional to the salinity
of the sea-water from which the ice has been formed. The first substance which begins
to separate at temperature below — 8-2°C is sodium sulphate (Na2S04). However, the
chlorine is retained since its separation begins only at — 23°C. This selective separation
process during freezing changes the composition of the salts in the sea-water (Ringer,
1906; see also O. Pettersson, 1883). Thus in the polar seas sulphate is expected to be
steadily withdrawn by the freezing process from the sea-water which thus becomes
enriched in chloride. On the other hand, in areas where the ice carried away by the
ocean currents melts sodium sulphate goes again into solution and the sea-water
should show a surplus in SO3.

Malmgren and Sverdrup (1929) have found that deviations of this type from
normal behaviour are only very slight, and thus there occurs no selective process on a
large extent during ice formation in nature. On the other hand, the investigations of
Liakionoff, according to Wiese (1938), have shown that in the Barents Sea both in
sea ice, as well as in melt water, there is a deficit of chloride and a surplus of sulphate
(SO3). Further investigation is required to settle this point.

(b) Density and Porosity of Sea Ice

The density of pure ice at 0°C is 0-91676, while the density of water at the same
temperature is 0-999867. The density of sea ice which is free of air bubbles in-
creases with its salinity. If it increases at the same rate as the density of sea-water
increases with salinity, the density of sea ice is expected to increase by about 0-0008
for every l%o in sahnity. The density of sea ice free of air bubbles and with a sahnity
of 15%o would thus be about 0-9296. The first precise determinations of the density
of sea ice were made by Makaroff (1901) by extensive measurements of the mean
height It and the mean depth d (above and below the sea surface) of freely floating
ice floes. If a^ is the density of the sea-water then the density of the sea ice is given by

This gives a mean value for the entire floe. Makaroff"'s measurements apply only to
summer floes of drift ice and give reliable values only for regular floes without any
snow cover. These observations gave results between 0-96 and 0-85. These large
variations are due to the considerable amounts of air and water which may be present
in sea ice. The greatest eff"ect is that due to air bubbles enclosed in the ice, which can
be of a twofold origin. One part originates already during the ice formation, due to a
separation of gases dissolved in the sea-water which cannot always escape from the
cells between the ice crystals. Thus the gas bubbles will be more numerous and larger
the faster the rate of freezing of the ice. The upper parts of freshly frozen ice thus
usually contain more air than the lower parts.

248

Ice in the Sea

A second source of air-bubble formation is the penetration of air during the meUing
process (Hamberg, 1895). In the upper part of a mass of ice which begins to melt
from the inside the rise in temperature first widens the small intermediate spaces con-
taining the salt solution. As the ice particles melt their volume decreases and empty
spaces are formed into which air is pressed in due to the atmospheric pressure. These
spaces finally become so enlarged that the melt water, together with the salt solution,
can flow out and finally they are replaced entirely by air. The originally pure and clear
ice thus becomes a porous mass penetrated by a number of air channels. On top the
drift ice floes in the summer thus always appear white simulating a snow cover.
The lower parts below the water surface are still cold and hard (solid). They do not
melt from the inside and show, at first, only very little porosity. However, when the
ice disintegrates more and more and still drifts in sea-water, the temperature of which
is above freezing point, the already existing empty spaces become filled with water
and the ice density increases rapidly.

The air enclosed in sea ice, according to Hamberg, has an oxygen content greater
than that of atmospheric air but less than that of the air mixture absorbed by sea-water
(24-26% as compared with 20-95 for atmospheric air and 34-6 for sea-water at 0°C
and 35%o salinity).

The most accurate determinations of the density of sea ice in situ have been made by
Malmgren (1927) on the "Maud" Expedition. These were made by determination of
the loss of weight of a piece of ice on immersion in petroleum of specific weight Pf
If the weight of the ice sample in air is G and in petroleum g grammes then the density is
given by

Pi =

Table 93. Density of sea ice

(According to Malmgren ("Maud" Expedition))

Depth of

Sample No.

Time

K°C)

Salinity

sample

Density

(%o)

(cm)

(g/cm^)

1-71

Jan. -Mar.

-26-4

0-919

-290

14-6

132

Max. 0-924

-220

3-6

Min. 0-914

8 and Sa^

Feb.

-240

0-921

Feb.

-290

1-9

0-918

10*

Mar.

-22-4

4-7

0-911

11^

Mar.

-270

0-857

12"

May

-6-2

—

0-885

13 and 14«

May

-6-2

—

0-892

^ Young ice partly broken open.

* Young ice from a freshwater pool on a thick old ice floe.
^ Young ice frozen in autumn from low salinity water.

* Thick broken young ice some time exposed to the sun.
^ Top peak of ice exposed to sun ("gesommert").

* Sample of old ice at the place of temperature measurement.

Ice in the Sea 249

Petroleum is particularly suitable as an immersion liquid because it cannot penetrate
into the small air-filled channels of the ice pieces. The results of these determinations
are summarized in Table 93, The conspicuous result is the very small variation in the
density of young ice, in spite of the strongly varying salinity of the samples and of the
equally variable depth from which they were taken, as well as of the changing thickness
of the floes. The smallest values (0-914 and 0-916) were given by two thin and highly
saline young ice floes which had been formed at very low temperatures. The rapid
freezing must of course have trapped a large number of air bubbles, probably more
than normal. The uniformity of the values between autumn and winter disappears
gradually in spring as melting becomes more and more eff"ective. There is a progressive
fall in density in late spring, and this decrease becomes stronger as the disintegration
of the ice proceeds during the summer. Values less than 0-90 show by the large number
of enclosed air bubbles that the ice must have been exposed to the sun ("gesommert").
The lowest value in density was found at the top peak of a large floe. During the pre-
ceding summer the salinity in this ice had been completely removed, and in winter the
melting water of it could be used for drinking water.

It is characteristic of sea ice that its thermal properties such as specific heat, latent
heat of melting and thermal expansion behave quite abnormally. During investiga-
tions of the heat expansion of sea ice Pettersson (1883) found that highly saline sea
ice expanded with decreasing temperature down to — 20°C, though for ice of lower
salinity this temperature was considerably higher. Malmgren showed by investiga-
tions during the "Maud" Expedition that Kriimmels' assumption, that this was due
to the salt solution enclosed in the ice, was correct. This abnormal behaviour rela-
tive to the specific heat, latent heat of melting and thermal expansion is thus also a
consequence of the formation and melting of pure ice occurring in the interior of
sea ice. At a temperature r, 1 g of sea ice of salinity l%o will contain a-r g of pure ice
and (1 — At) g of salt solution. If the specific heat of sea ice at the temperature t
is Ct then this quantity of ice for a temperature change dr will require a quantity of
heat Crdr. It is made up essentially of: (1) the rise in temperature Orcdr of pure ice
(with specific heat c); (2) the rise in temperature of the salt solution (1 — a-^Kdr
(with specific heat k); and (3) the heat Kda^ required to melt da-r g of ice (with latent
heat of melting A^). This gives the equation

Cr = a-rC + (1 — a-)K -f Xr -^ .

dr
Since the second term is small and as a first approximation a^ = 1 then

Ct = c + A, — -.
dr

For sea ice of salinity 5'%o the variable amount of ice is Sda-r and therefore one ob-
tains for it the relation:

c.^c^-Sx/--^.

250 Ice in the Sea

According to p. 247 if Sr is the salinity of the salt solution

(1 - ar)S, = 1,

so that

da, 1 dSr

and

Cr = C + Xr

S2 dr '

S dSr

5? dt '

According to the investigations of Pettersson (1878) A^ = 80 + 0-5t, The factor of
Xj can be calculated from investigations made by Ringer, so it is therefore possible to
evaluate the above equation for different temperatures and salinities (Table 94).

Table 94. The specific heat of sea ice
(According to Malmgren)

Temp.(°C)

-2

-4

-6

-8

-10

-12

-14

-16

-18

-20

-22

r 2

2-57

100

0-73

0-63

0-57

0-55

0-54

0-53

0-52

4-63

1-50

0-96

0-76

0-64

0-59

0-57

0-56

0-55

0-54

5'%o «

6-70

1-99

1-20

0-88

0-71

0-64

0-61

0-60

0-58

0-57

0-56

8-76

2-49

1-43

101

0-78

0-68

0-64

0-61

0-60

0-58

10-83

2-99

1-66

0-85

0-73

0-68

0-67

0-64

0-62

0-60

ll5

1601

4-24

2-24

1-46

102

0-85

0-77

0-76

0-71

0-68

0-65

Malmgren has also determined the specific heat of ice samples experimentally,
and has obtained values in excellent agreement with the theoretical. At higher tem-
peratures the heat capacity of sea ice is quite high, at — 2°C and 15%o salinity it reaches
16-0 g cal. These high values can be explained either by melting or freezing of large
amounts of pure ice in the salt cells of the ice at temperatures close to freezing point
and fjr temperature changes of about 1 °C, which is accompanied by release or uptake
of large amounts of heat from the latent heat of melting. For sea ice the specific heat
and the latent heat of melting are properties closely related to each other.

The dependence of the latent heat of melting on temperature and salinity can also
be calculated theoretically from Sr the salinity of the ice, and r^, the freezing tempera-
ture of sea-water of salinity S. If t is close to zero, the latent heat of melting for pure
ice will be constant between r and r^ and will be 80 g cal. The amount of heat required
to melt 1 g of sea ice will be made up of: (1) the heat = 80[1 — ^(1 — a-r)] required
to melt pure ice; and (2) the heat required to raise the temperature of the pure ice
and the salt solution from r to Tj,. Since the specific heat of pure water is 0-5 this
quantity of heat will be approximately 0-5 (xg — T)aT. The latent heat of melting of
sea ice will thus be given by

= ^°('-|)

+ 0-5(t. - t)

Ice in the Sea
Table 95. Latent heat of melting of sea ice

251

Salinity (%o)

2 4

„ r-io°c

80
81

72
77

55
68

46
63

37
59

16
19

Table 95 shows values for different salinities and for temperatures equal to 1° and
-2°C.

The coefficient of thermal expansion can be calculated in a similar way; it
is made up of the coefficient of thermal expansion of pure water (a = 1-7 x 10"'*)
and a term which depends on the amount of ice forming or melting due to the change
in temperature in 1 cm^ of sea ice. Since the freezing of 1 g of water at t° is accom-
panied by an increase in volume of yr = 0-091, the coefficient of thermal expansion
of sea ice will, according to the above discussion, be given by

dttr S dSr

Table 96. Coefficient of thermal expansion of sea ice (Ur X 10'*)

(According to Malmgren)

Temp. (°C) . .

-2

-4

-6

-8

-10

-12

-14

-16

-18

-20

-22

4
6
8
10
.'5

-22-10
-45-89
-69-67
-93-46
-117-25
-176-72

-4-12
-9-92
-15-73
-21-53
-27-34
-41-85

-1-06
-3-81
-6-55
-9-30
-12-05
-18-92

016

0-83

1-13
0-56
0-00

1-23
0-78
0-33

1-27
0-85
0-43
0-02

1-33
0-96
0-60
0-23

1-38
1-07
0-76
0-45
014

1-44

Salinity %„

-1-37
-2-90
-4-43
-5-95
-9-78

-0-02
-0-88
-1-73
-2-59
-4-73

1-18
0-93

-U-D7
-M3

-2-54

-0-13
-0-59
-1-72

0-67

-U-40
-1-45

-0-13
-1-03

0-42

-0-63

-0-22

From this equation Malmgren has calculated the values given in Table 96, and experi-
mental determinations of Ur, on samples of natural ice have fully confirmed the theor-
etical values. There is an essential difference between Uj for sea ice and freshwater ice.

Pure ice always expands with increasing temperature; sea ice expands only to a
lesser extent, and then only at very low temperatures and low salinities. Thus the
second term in the above equation becomes unimportant. At higher temperatures and
salinities the second term predominates; this means that the ice volume increases
with decreasing temperature and at very low temperatures and high salinities this
increase may be considerable.

Extensive series of temperature recordings at different depths in sea ice (ice floes)
have been made by the "Fram" Expedition 1893-6 and the "Maud" Expedition
1918-25. The latter were obtained by using electrical resistance thermometers and are
much more reliable. Table 97 gives monthly means for five depths down to 2 m for
every month during which the snow cover at the place of measurement was left
undisturbed.

The annual temperature variation at all depths can be approximated closely by a
simple sine curve of the form

Af + a sin

/ + a

252

Ice in the Sea

and the result of this analysis, given in the last lines of Table 97, shows how regular
is the annual temperature wave, with a decrease in amplitude and a phase shift in the

Table 97. Annual temperature variation at different depths in sea ice
(According to the values of the "Maud" Expedition, North Siberian Shelf)

Depth (m)

000

0-25

0-75

1-25

200

Jan.

-280

-24-1

-18-9

-140

-6-5

Feb.

-30-9

-26-9

-21-3

-16-3

-8-5

Mar.

-291

-26-0

-21-0

-16-5

-9-6

Apr.

-21-6

-20-1

-17-3

-14-4

-9-4

May

-7-4

-8-6

-9-3

-9-2

-7-4

June

-1-5

-30

-4-1

-4-5

-3-8

July

-00

-01

-1-3

-1-7

-1-8

Aug.

-00

-0-8

-11

-1-2

Sept.

-4-7

-1-3

-0-9

-11

-1-3

Oct.

-12-3

-7-6

-3-3

-1-6

-1-4

Nov.

-23 0

-17-8

-11-9

-7-1

-2-4

Dec.

-29-9

-24-4

-17-7

-12-2

-4-6

Mean M

-15-70

-13-32

-10-65

-8-31

-4-82

aCC)

16-82

14-60

11-17

8-36

4-40

a (degrees)

259-6

250-9

240-9

230-7

210-4

extremes, penetrating into the ice (Fig. 112). In both series of recordings there is good
agreement in the upper layers of ice down to about 1 -5 m, but this is not true at greater
depths. The "Fram" values are too low, probably due to the observational method
using bar-thermometers. The decrease in the annual amplitude with depth shows the
same. According to the "Maud" values the annual variation disappears at a depth of
2-9 m. At a depth of 2-8 m the temperature of sea-water underneath the ice floe
reaches — 1-6°C and remains constant throughout the whole year. At the side under-
neath an ice floe, the thickness of which varies on the average as seen from Table 97,
the amplitude of the annual temperature variation thus falls to zero.

Fundamental investigations on the thermal conductivity of ice have also been made
by Malmgren. Stefan (1890) found a thermal conductivity coefficient k = 4-3 x 10~^
from theoretical investigation of the process of ice formation, but this value can only
apply for freshly formed pure ice. Later Mohn (1 900) attempted to compute the thermal
conductivity coefficient from the decrease in the annual temperature variation and
from the retardation of the extremes with depth in ice ffoes. However, these methods
cannot give reliable values since the theory is valid only for infinite thickness, while the
thickness of sea ice is small and the lower side of a floe remains almost always at a
temperature of — I-6°C. Correct values of A' can be determined, according to Malm-
gren, from the temperature gradient and its change with time at diff'erent depths.
Assuming a cylinder with a vertical axis through an ice floe, then definite amounts of
heat will enter the cylinder through its upper surface in / sec. If the ice floe is of suflH-
cient horizontal extent no heat will pass through the vertical wall of the cylinder and
the heat flux will only occur normal to the surface of the ice floe. If the heat content
of the cylinder for a given time remains constant then kiGi = k^G^, where Ati, k^ and

Ice in the Sea

253

0 00.

^025

2 00

1 n E Ez: Y 3a ^zasnux x xixn
Fig. 112. Annual temperature variation at different depths in sea ice.

Gi, Gz are the thermal conductivity coefficients and the temperature gradients at the
upper and the lower surface of the cylinder. However, if the mean temperature changes
from Tj to T2, then the following relation holds:

(ArjCi — koG^t = hco{r^ — Tg),

where h is the height of the cylinder, c the mean specific heat and a the mean density.
From observations of temperature in ice it is possible to find cases where the mean
temperature of a layer is constant for a certain time, and cases where it undergoes large
rapid changes. The above equation can then be used to calculate A^ and k^. Table 98
shows numerical values for k determined for the winter periods 1922-3 and 1923-4.
They are of the same order of magnitude as the mean values obtained by Stefan but
have a marked dependence on the depth (Fig. 113).

Table 98. Thermal conductivity of sea ice at dijferent depths
(According to Malmgren)

Depth (m)

60 and 75
resp.

125

„,. , / 1922-3 2-4

Winter |j923^ 1-7

3-6 40
3-3 4-5

4-2 X 10-3
50 X 10-3

There is a rapid decrease in the thermal conductivity in the top layers of sea ice
which must be due to the numerous air bubbles in these layers (density about 0-88).

254

Ice in the Sea

Deeper in the ice the thermal conductivity approaches a limiting value of 5-0 x 10~^
which corresponds to the value obtained for clear freshwater ice without air bubbles.
Malmgren's determination of the physical constants of sea ice are of considerable
importance in questions of the heat balance in polar regions, since they allow the de-
termination of the amount of heat gained by the surface of the ice in polar regions and

%
o

0 I 2

Fig. 113. Changes in thermal conductivity in sea ice with depth.

thus also by the atmosphere immediately above it from the water below. For the greater
part of the year the water underneath is warmer than the ice cover and the air above it,
and therefore there is a continuous flux of heat upwards. Such a calculation can be
made with the temperature observations of the "Maud" over a period of a year.
The total amount of heat passing through the different depth-levels in a year amounts
on the average to 6800 g cal/cm^ and should be the same for all levels. This amount
of heat is released to the atmosphere above the ice year after year. In the cold season
of the year when the temperature gradient is several times larger this flux of heat is
greater; in summer it may even be reversed but is then never very large. Taking a
depth of 0-75 m as representative for the entire layer of ice, the amount of heat,
W^o-75 passing through this level per cm^ and month can be calculated, knowing the
temperature gradient for each month during the colder season of the year. Part of this
heat serves to raise the temperature of the 0-75 m thick surface layer. If the tempera-
ture difference between the beginning and the end of the month is Jr then the heat
gained by the atmosphere during that month is

= f^o-75 - IScaAr = fFo.75 - 34-4 J T.

The values calculated by Malmgren using this equation for the months from Septem-
ber 1923 to April 1924 show that during the cold season of the year the atmosphere
receives the very large amount of 76,700 kg cal/cm^, which is sufiicient to melt 96 cm
of ice. However, large as this may appear, it is only a ninth part of the heat that the
European Mediterranean, for example, provides to the atmosphere (676,000 kg
cal/m^). However, in the polar regions its eff'ect is none the less still important. During
the cold part of the year there is a thin layer of cold air over the Polar Sea, extending
to a height of about 150 m (Sverdrup, 1926). This layer of air has such a stable
stratification that it mixes only to a very small extent with the air above. The heat from
below is thus imparted almost entirely to this layer and prevents a decrease of the

Ice in the Sea 255

temperature to very low values. The increase in temperature per day due to the flow
of heat Wa from below can be found from the mean height of this cold atmospheric
surface layer. As Malmgren showed, this heat is quite large and it is obviously this
source of heat that prevents an intensive cooling of the atmosphere above the North
Polar basin. The temperature can thus never reach the low values found in central
Siberia or central Greenland, where this heat source is not available.

{d) The Mechanical Properties of Sea Ice

The continuous formation of ice by freezing is counter-balanced by very effective
processes that reform and destroy the ice fields. The mechanical properties of ice
(elasticity, plasticity and resistance against deformation, bending and compression)
are of the greatest importance in the interplay between these processes. Large ice
surfaces seldom remain unchanged for longer periods. They are broken up rapidly
from the edges, by the combined action of the wind, waves and periodic tidal currents,
and in a short time become separate ice floes. With the aid of strong winds they are
piled up by the large horizontal pressures and pushed one above the other. The
resultant mass, when finally covered with snow, cemented together and built up into
several layers, is pack ice. Pressure and tensions are common in the polar regions
(especially in the Arctic). Gaps and open spaces may exist for a short time but are
rapidly covered over by young ice which again re-unites the whole mass. These
pressures are not due to the effect of the wind alone, because often the wind only
influences far-off regions, thereby subsequently causing pressures in the Arctic (distant
effect) ; they are often due to rapid temperature changes at the surface of the ice. Since
the under-side of an ice floe is always at the temperature of the water (near freezing
point) there will be tensions and stresses in the floe. Figure 1 14 shows schematically
the cracks and fissures formed when the stresses due to thermal expansion at the surface
exceed the elastic limit. In the same way thermal contraction at the surface forms in an

Fig. 114. Changes in an ice floe due to thermally induced expansion.

analogous manner cracks at the lower side. The cracks on the upper surface of the
floe soon fill with snow and melt water and those in the bottom surface fill with ice
due to the rapid freezing of sea water in contact with the cold ice. There is thus a
continuous formation of ice. The ice-covered regions in the Antarctic are not basins
surrounded by land, and therefore ice pressures occur less often and are considerably
weaker. The humps, hummocks and ridges of piled-up floes, known by the Siberian
name toross, which are sometimes up to 5 m or more in height are much less common
in the Antarctic pack ice; instead the action of pressure often forms folds and flexures.
The mechanical properties of ice, like its other properties, depend on the temperature
and salinity, but due to the multiplicity of ice forms and conditions these determine
only the order of magnitude, and there may be considerable variations caused by the

256 Ice in the Sea

special structure of an ice floe and its past history. The most important of the mechani-
cal properties is the elasticity, which is characterized by Young's modulus E and the
modulus of rigidity /x. Ice is of course composed of ice crystals and its elasticity is not
the same in all stress directions. An ice crystal can be regarded as built up of a large
number of thin platelets at right angles to the crystal axis. Deformation at right angles
to this axis meets a much smaller resistance than one in the direction of the axis.
The different values for Young's modulus shown by different natural samples are
probably due to this. Few direct determinations have been made of the elasticity
constants for sea ice, but they have been determined more often for fresh water ice by
a variety of different methods. The more reliable values for the modulus of elasticity
E are those of Reusch, which give 23, 632 kg/cm^. Its variability with the position
of the crystal axis relative to the axis of force has also been determined, giving between
18, 200 and 38, 300 kg/cm^. E increases with decreasing temperature.

A more accurate determination of these constants can probably be made indirectly
by measurement of the velocity of elastic waves in the ice, and a large number of de-
terminations of this type have been made. Ewing, Gray and Thorne (1934) measured
this velocity in thin ice rods and found the following values for the elasticity constants:

Young's modulus E Rigidity modulus fi. Poisson constant a

9-17 X IQio dyn/cm^ 3.36x lO^odyn/cm^ 0-365

Seismic measurements of the thickness of the ice on alpine glaciers and in Greenland
(Brockamp and Mothes, 1930) have given

E = 6-82 X lO^o dyn/cm2; ^i = 2-51 X lO^" dyn/cm^; a = 0-361.

Considering the difference between experimental and natural conditions these values
agree quite well. The elastic limit in ice is not large; for river ice Weinberg found
0-57 kg/cm^; for granular glacier ice Hess found 0-09 kg/cm^. The plastic limit is,
of course, much higher.

The strength of ice of different origins provides a more useful comparison than the
above numerical values and has been used by Makaroff. His measurements show
clearly that freshwater ice is of much greater strength than sea ice and that an in-
creasing salinity in the water in which it is formed and a higher temperature, makes
the sea ice less resistant. Weinberg (1907) investigated the strength of a large number
of sea-ice samples and found that the values obtained usually increased with decreas-
ing temperature; compared with the values at — 3°C there were increases of 20%,
35% and 45% at -10°, -20° and -30°C respectively.

Investigations of the deformation of ice under the effect of continuous pressure have
been made by Andrews, and especially by Royen (1922). From their results, it is worth
mentioning that the plastic deformation of ice under the influence of continuous
pressure can be expressed by the equation

pi^T

1 - T

where p is the pressure (load) in kg/cm^, T is the duration of this pressure in hours, t
is the mean temperature of the ice and k is a constant characteristic for each sample and

Ice in the Sea 257

varies within the hmits 6 x 10-' and 9 X 10"*. These investigations showed the con-
siderable effect of the temperature on the hardness of the ice. The strength of ice is
very important in calculating the loads that can be put upon it. The following empirical
data may be given based on experience : freshwater ice 4 cm thick will carry a man,
from 10-12 cm thick a galloping horse, from 15 cm thick a heavy-loaded truck, and
over 45 cm thick a railway train. This question is also of importance for aircraft
landing on ice. Moskatov (see "Die Naturverhaltnisse des Sibirischen Seeweges"
("Conditions along the Siberian Sea route"), Oberkom. Kriegsmarine, BerUn 1949,
p. 84) has given the following table for the minimum safety thickness of freshwater
ice for aircraft landings :

Aircraft weight (tons)
Minimum thickness (cm)

2
15

5
24

15 20

39 45

The strength of sea ice, and that of salt-free ice formed from sea ice due to a decaying
process of several years is considerably less than that of freshwater ice. To carry
the same load the ice in the centre of the Arctic basin must be two to three times
thicker.

3. Ice Conditions and their Seasonal and Aperiodic Variations in Arctic and Ant-
arctic Regions

(a) Ice Conditions of both Polar Caps

In the Northern Hemisphere sea ice is largely confined to the Arctic Mediterranean,
the central basin of which is always covered by it. Figure 115 shows the general out-
lines of mean ice coverage in summer and winter (Budel, 1943, 1950). September is the
time of minimum extension in ice cover, and the ice is limited to the inner part of the
North Polar Basin, which at that time is most remote from the warm land masses.
This ice lasts throughout the summer and then extends again enormously during the
winter. Except in the area of Gulf Stream water it reaches everywhere to the northern
coasts of the continents and extends as long tongues of pack ice along the eastern
coasts of Greenland and Labrador. To this winter ice then adds the one-year-old
winter ice of the adjacent seas. In winter, of the total area of the North Polar Basin
(11-6 milhon km^) on an average 8-7 miUion km^, (or 75%) are covered by ice. If
the pole were surrounded by land with a circular area of 2-9 million km^ then the
above mentioned ice-coverage would extend southward everywhere to the 72-7°
parallel (thus everywhere 17-3° lat. distance from the pole).

In the Southern Hemisphere, where the Antarctic land mass surrounds the South-
pole with a total area of 14-8 million km^, the ice-coverage is 29-0 million km^ and
for an even distribution would then reach northward to the 55-8° parallel. These
figures show the strong contrast in ice conditions between the two polar regions.
The ice covers 3-35% of the total Northern Hemisphere, but 11-30% of the total
Southern Hemisphere.

In the Southern Hemisphere (see Fig, 1 1 6) the ice extends uniformly around the
central Antarctic continent, enclosing it on all sides, and the symmetric circumpolar
arrangement of the ocean surface and the ocean currents fix zonal drift ice limits

258

Ice in the Sea

Fig. 115. Average extent of sea ice (mean drift ice limit) in the Northern Hemisphere for

summer and winter:

mm^

AAA

m///m

°o°o°

AVERAGE DISTRIBUTION OF

Polar ice coverage closed in summer (about beginning of September)
Brocken polar ice coverage in summer (about beginning of September)
Southernmost iceberg limit in summer (May to September)
Closed polar ice coverage in winter (March to April)
Brocken polar ice coverage in winter (March to April)
Southernmost iceberg limit in winter (October to March)
Closed ice on inland seas and lakes in winter (February to March)
Brocken ice on inland seas and lakes in winter (February to March)

without any large meridional irregularities. In the Northern Hemisphere, on the
other hand, the continents and the eccentrical position of the large polar icelands
confine the ice field on all sides, and allow warm ocean currents to enter at only one
gate, between Iceland and Scandinavia where the warm Atlantic current pushes the
limits of drift ice back to the northern coast of Spitzbergen and into the inner parts
of the Barents Sea.

Ice in the Sea

259

Referring to the special regional distribution of the three different types of ice (polar
ice, pack ice and solid ice) the central area of the North Polar ice consists always of
pure polar ice (Smith, 1931); it is 3-3-5 m thick at the end of the winter and 2-2-5 ra
thick at the end of the summer. It covers about 70% of the entire Polar Basin, i.e.
5-2 million km^. It is usually a continuous layer, but especially towards the edges it is
split up by ice pressure into large ice fields and ice floes. This large polar ice cap is
closely confined to the 1000-800 m isobath and has a more or less elliptical shape
lying much nearer to the continental coast and coastal islands on the Greenland-
North American side than towards the coast between Spitzbergen and Alaska where
the broad Siberian Shelf lies between. The centre of the polar cap is often called the
"pole of inacessibihty" and is situated about 400 nautical miles north of Alaska.

The maintenance of this polar ice cap represents a state of equilibrium with the total
annual growth. The total gain consists at first of an addition of ice from the surround-
ing pack ice zone due to freezing at the bottom layers of ice floes, secondly of snow
falls on the ice surface and the re-freezing of open spaces. The ice loss is caused by

Northern limit of drifting fiores <i

-- Northern limit of ice bergs

Pack ice limit

Fig. 116. Average extent of sea ice in the Southern Hemisphere; the dotted line .... gives

the mean northern iceberg limit, the continuous line gives the northern limit of pack

ice and the broken line the northern limit of drift ice.

260 Ice in the Sea

evaporation, melting and the southward drift of ice away from the edges. It can
reasonably be assumed that the equilibrium is of a quasi-stationary nature.

Our knowledge of the movements of the polar ice is largely obtained from the drift
of vessels beset in, i.e. frozen into, the ice. These show that, at least in the western
half of the polar ice cap, there is an east-west drift ; on the North American side there
appears to be a drift in the opposite direction so that in the North Polar Basin there
is a general anticyclonic ice drift. North of Greenland and Grant's Land, however,
the drift is directed towards the area between Greenland and Spitzbergen. The speed
of the ice drift to the north of Franz Josef Land and towards Spitzbergen is about 1
nautical mile a day; the "Sedow" found values twice as great; the "Maud" found
values between 0-6 and 3-2 nautical miles a day; the "drifting polar station" found at
first, near the pole about 4 nautical miles a day and then, after a decrease to 2-4
nautical miles a day, a further increase to 5-6 off the Greenland coast. Similar values
have also been found by means of drifting buoys which have been laid out recently
inside the North Polar basin by the Russians.

The pack ice zone is continuous with the polar ice zone and covers about 25% of
the North Polar Basin ; in summer it usually forms the southern limit of the drift ice
fields, here broken up by kilometre-long channels. The part over deep water pro-
ceeds with the motion of the ice drift though probably at a lower speed, but in shallow
waters (over the shelf) its movement is towards the east. As a consequence of this
opposite movement, the ice fields in the intermediate areas are very much broken up
and large, and sometimes navigable, fracture zones appear (termed "polynya" by
Russian research workers). The main fracture zones run north of Spitzbergen, Franz-
Josef Land, Sevemaya Semlja the New Siberian Islands and Wrangel Island. They are
particularly well marked to the north of these islands and may occur even in winter
during persistent south-easterly and southerly winds. The pack ice penetrates extremely
far southwards into the North Atlantic in two places; (1) along the east coast of
Greenland until Cape Farewell and around it ; (2) along the eastern coast of North
America from Baffin Bay southwards in the Labrador current as far as the Grand
Banks of Newfoundland. These ice currents carry not only pack ice from the North
Polar Basin but also winter ice and solid ice from the Greenland Sea and from the
northern part of the Baffin Sea. In both outflows there is an outer zone of drifting ice
floes, a middle zone of more compact ice with occasional channels running through it
and finally an inner core of solid ice joining the solid ice along the coast. Smith gives
the following data (Table 99) for these two ice currents.

The pack ice zone is bordered by a zone of solid ice which transforms into the land
in coastal areas. During the winter in both Northern Siberia and in the North Ameri-
can Archipelago it covers all channels, bays and fiords, etc., and these only become
free of ice again in summer. In coastal areas the solid ice at the beginning of the sum-
mer contains earthy material (stones and shells) picked up by freezing of ice of the
sea bottom melting out in sunraier; the surface of the ice is then often brownish
(Transche, 1928).

In the Antarctic (Drygalski, 1921) floe ice occurs only outside a certain broad belt
containing icebergs and the remains of icebergs. This belt extends for the most part
to about 60° S. but reaches farther north near the Falkland Islands and South Georgia
and past 50° S. only near Bouvet Island. The ice floes are frequently found in large

Ice in the Sea
Table 99

261

Total drift

time of a

Ice current

From

Distance

(nautical

miles)

Mean speed

(nautical

miles/day)

single ice
field from
origin to
end given
in months

East Greenland ice

75° N., 0°W.

62°N., srw.

1850

7-5

East American ice

(Baffin and
Labrador ice)

74° N., 70° W.

45°N., 49°W.

1950

12-5

groups, sometimes associated with icebergs which were formerly frozen into them.
South of 64° S. begins the continuous drift ice which is held together by the westward
directed currents which tend to the south due to the influence of the Coriolis force.
Icebergs are more frequent here and have the table-form characteristic of the Antarctic.
The ice masses are driven together partly by the wind, but the ice pressure is not as
strong here as in the Arctic, The belt of drift ice extends to the edge of the continental
shelf.

In the shelf zone over the shallow waters the ice is a mixture of floating ice floes and
icebergs which form here and accumulate. The whole mass is held together by the
larger icebergs stranded in shallow water. Superficially the shelf ice appears as a
flattened, smooth, rounded ice-surface because of the frequent snow storms, but if
the upper parts of the ice is broken off by the wind, the solid ice layers stand out more
clearly; these have been termed blue ice by Drygalski on account of their colour. This
permanent region of shelf ice between the drift ice and the coast completely surrounds
the coast and obhterates the actual coast-line. This is the main cause of the uncertain
charting of the Antarctic continent.

There are only a few approximate estimates of the budget of ice transport of the
total polar regions. Krummel (1907, p. 515) gave the following approximate summary
for the Northern Hemisphere: between Spitzbergen and Greenland the main carrier
of outflowing sea ice is the East Greenland Current. It has a width of about 500 km,
and according to Makaroff" in summer and winter about 76% of its surface is covered
with ice floes and pack ice. Taking the mean velocity of the current as about 10
nautical miles a day and the average thickness of the ice as 5 m then the annual
volume of ice carried out from the central Polar Basin by the East Greenland Current
will be 12,700 km^. This is about one-third of the total pack ice and polar ice in
the entire North Polar Basin. Another stream of ice floes comes from Baffin Bay.
This has a width of 200 km when leaving Davis Strait and on the same basis as before
will carry somewhat more than 5000 km=^ a year. If, furthermore, the drift ice entering
the Barents Sea is estimated as 2000 km^, there must be a total annual flow of about
20,000 km^ of ice to be melted in the northern part of the North Atlantic each year.

{b) Seasonal Displacements of the Ice Limits

The inner part of the North Polar Basin is covered by polar ice throughout the year
and changes appear only at the edges in the outer ice zone, especially near land areas.

262

Ice in the Sea

263

The solid ice of the coastal shelf shows particularly a pronounced annual variation
since it melts away almost completely during summer and re-forms again during the
autumn. The North European and Siberian Shelf areas thus show large seasonal
displacements in the ice limits. In the eastern part of the Siberian sea-way east of
Novaya Zembla and remote from the influence of the North Atlantic current the
distribution of ice, even in the summer months, may change so rapidly and so much
that it is difficult to give exact mean ice limits for individual months (see Atlas der
Deutschen Seewarte, 1942; BiJDEL, 1950; Nusser, 1952).

In the Barents Sea, which is particularly influenced by the Atlantic Current, the
seasonal displacements of the ice limit are very large. The two small charts in Fig.

117 present the mean ice limits as separate monthly means for a 10-year period from
1929 to 1938. One of them (March to August) shows the retreat of the ice limit in
spring and summer, and the other (September to February) shows its ad\ance in
autumn and winter. During tliis period from 1929 to 1938 ice conditions were par-
ticularly favourable and this should be borne in mind.

The monthly limits of the ice along the eastern coast of Greenland, in the Davis
Strait, in the Baffin Bay and along the east coast of North America as far as the Grand
Banks of Newfoundland are almost entirely within the region of inffuence of the two
great polar currents, the East Greenland Current and the Labrador Current. Figure

118 shows two charts, again for the period of retreat (March to September) and ad-
vance (September to February) (see also Atlas der Deutschen Seewarte, 1940, means
for the years 1929-38). The east coast of Greenland is blocked for almost the whole
year by a belt of ice varying strongly with latitude ; this coast is only free of ice in the

^ 50° 40° 60° Vl/

Fig. 118. Average ice limit along the eastern coast of Greenland, in Davis Strait and Baflfin
Bay, and along the east coast of North America for each month (1929-38).

264

Ice in the Sea

southernmost parts in ice-poor years and frequently, even in the summer, there is a
broad belt of drift ice off the south-west coast (Julinaehaab), although the fiords are
completely ice-free.

Iceland is usually entirely ice-free, but during steady northerly winds ice may be
driven against the north-west coast, drifting then along the northern coast towards
the east where, in unfavourable years, this ice may be united with the drift ice moving
towards the south-east in the East Iceland Current. The north coast of Iceland is then
partly or entirely blocked by the ice. The probability of ice occurrence off the coast of
Iceland is not small, as Table 100 shows (Meinardus, 1906; Brooks and Quennel,
1928), The maximum ice season around Iceland is in early spring (March and April).
At this time ice is observed about every second year off the coast and usually remains
there nearly a full month.

Table 100. Frequency

and persistence of ice occurrence around Iceland 1801-1900

Month

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Probability of ice in (%)

Average duration given
in % of each month

24
14

28
20

|31

56t
46t

22
18

14
9

4
1

1
1

3
1

5
1

t Maximum

The entrance into the Davis Strait and into Baffin Bay is completely blocked from
October to November ; however, beginning in April the eastern part of the northern
Baffin Sea becomes ice-free in the east. In the central parts an ice-free area forms al-
ready in May, growing in extent during the following month and joining with the open
water on the eastern side during July. This phenomenon is due to currents carrying
warmer Atlantic water into the Baffin Bay weakening and breaking up the ice fields
from beneath. Another further peculiarity is the formation of the "middle pack" a
mass of ice surrounded by open water that still occurs in the western part of Baffin
Bay during August and September,

The drift of pack ice along the coasts of Labrador and Newfoundland begins in late
autumn (October and November), and reaches the northern parts of the Grand Banks
of Newfoundland in January or, at the latest, in February. The ice fields reach their
greatest extent, though not their greatest intensity, during April and the ice limit then
begins to retreat. Figure 1 19, according to Huntsman (1930), presents the extent of the
pack ice south of Newfoundland and shows clearly the position of the main ice-
fields relative to the Labrador current system and to the Gulf Stream flowing farther
south.

Mecking (1906, 1907) made a detailed investigation of the dependence of the ice
drift from the Baffin Bay on currents and weather. Table 101 shows mean values for
a period of 1 8 years of the monthly amount of ice presented as a percentage of the
annual amount of ice in the area of the Grand Banks of Newfoundland.

The season with ice fields lasts from January to about the middle of July ; then it
ends rapidly in August, when rapid melting due to higher air and water temperatures
occurs. The secondary maximum in May is due to the icebergs, which are also at a
maximum at this time (Fig. 1 20).

Ice in the Sea

265

Table 101. Mean monthly ice amounts (as a percentage of the mean annual ice) for the

Newfoundland Grand Banks

Month ! Jan.

Feb. Mar.

Apr.

May June July

Aug.

Sept.

Oct.

Nov. Dec.

Drift ice or
ice fields 9

37t 1 18

14t 5

0 0

(t Max., I secondary Max.)

In the Pacific Ocean, sea ice is limited to the north-western marginal seas; Bering
Sea, the Okhotsk Sea and the Sea of Japan. The ice limits for the periods of advance
(November to March) and retreat (March to July) are shown in Fig. 121. In the north
the Bering Sea is connected through the Bering Strait with the Tschuktschen Sea
where ice is plentiful. The mean annual duration of ice here is about 270 days, and the

W 70°

Fig. 119. Chart of the distribution of pack ice south of Newfoundland (according to
Huntsman). The short thick lines show the position of the ice fields at the time of maximum

ice extent.

Fig. 120. Annual ice variation in the area of the Newfoundland Banks. The full curve shows
the relative volume of drift ice normally present south of Newfoundland; the dashed curve
shows the mean number of icebergs present south of Newfoundland and in the western
Atlantic. The lower curve gives the mean number of icebergs south of the Grand Banks.

266

Ice in the Sea

strait is only completely ice-free from July to the end of September (Hegemann,
1890; ScHULZ, 1911). To the south and south-west of the Lorenz Island along the
east coast of Asia the ice season is still very long; in the Tartar Gulf (between the con-
tinent and Sakhalin, 52° N.) it lasts through half the year and in the Gulf of Vladi-
vostok (43° N.) 3 months. The climatic conditions of the neighbouring continent with
its monsoon-like winds blowing oft' the land carry a strong continental type of cU-
mate well out over the ocean, and even in spring when the land already warms up,
the cold ocean currents along the coast prevent the break-up of the ice. Ice is still
present in the inner parts of the Okhotsk Sea in May and the last only disappears in

120° E

70° W

Fig. 121. Ice limits for the months from October to July in the north-western adjacent seas

of the Pacific Ocean.

Ice in the Sea

267

July. At the end of October, and during the first half of November, ice formation
begins again along the northern coast. Conditions here are quite different from those
along the east coast of North America and Greenland since the ice masses in these
adjacent seas of eastern Asia are always of local origin, and are not reinforced by
Arctic pack ice and icebergs as in the East Greenland and Labrador Currents.

Knowledge of the annual variations of the ice coverage in the ocean surrounding
the Antarctic is still very poor. The ice limits in each month have only been known with
some accuracy since the intensification of whaling. The pack ice limits in the Ant-
arctic between 40° W. and 1 10° E. have been given by Hansen (1934) for the 4 years
from 1929 to 1934 (Fig. 122). In the area east of South Georgia to about 20° E. the

Fig. 122. Pack ice limits in the antarctic region between 40° W., and 1 10° E., for the whaling
season 1930-1 (according to Hansen).

ice extends to a latitude of about 55° S. at the beginning of November. Deviations
from this value in the course of the year are small. East of 20° E. the limit trends
more and more to the south. As the season advances this outer pack ice retreats
slowly towards the south, and by December whaling ships can penetrate it and reach
open water at about 60° S. There is then usually no pack ice until the inner pack
ice coast is reached.

These two ice zones, the inner drifting westwards and the outer eastwards, are
characteristic for the whole region from the Weddell Sea as far as 20-30° E. They are
about 7° lat. apart. There is no such subdivision in the ice drifts east of Enderby Land.
The outer pack ice zone melts very rapidly, especially if it is broken up in a number of
places into large drifting ice masses. In the region of the Antarctic Ocean from
Enderby Island and Balleny Island the ice limit retreats steadily during the melting
period; the oceanic currents directed northward (equatorward) in 60° S. carry the
ice floes in the same direction whereupon they rapidly disperse and melt.

268

Ice in the Sea

It is not surprising that a natural phenomenon such as the polar ice coverage de-
pending on such a large number of different factors should show large aperiodic
variations. In addition to their scientific interest these variations are of considerable
practical importance for life and commerce in the polar regions. Statistics of the
changes in ice coverage in the polar seas do not go very far back.

Following a resolution of the seventh international meeting of geographers, Berlin,
1899, the Royal Danish Meteorological Institute has published since 1894 an annual
ice-record for the Arctic, and these annual reports are now the most important source
of data of this type. However, knowledge of the extent and movement of the ice is
confined mostly to shipping routes and fishing areas. The available data are thus in-
homogeneous and incomplete. However, more accurate observations of the polar
zones from the air will probably lead in the future to further progress, especially also
because of the increasing military interest.

Variations in the ice conditions of polar and subpolar regions do not proceed every-
where in the same way; because they appear to be due, in the first place, to variations
in the atmospheric and oceanic circulation, both of which regionally cause quite
different effects. Somewhat more detailed investigations of these aperiodic variations
have been made for the oceanic regions around Iceland, Davis Strait and Newfound-
land and also, in part, for the Barents Sea.

Meinardus (1906) has examined the duration and the intensity of ice in the area
around Iceland for the years 1800 to 1904. Figure 123 shows that Iceland is situated on

(a)

I I il

A/^

-, i-i^d

\ h

JMu^

7tl"yu\/iiAiil

(b)

h\\

.iSMfjAa

\.K

I'VI

1«lf V

\ ip

V^vyv^

1800

1820

1840

I860
Years

1880

1900

1920

Fig. 123. {a) Character of the ice-years around Iceland for the period 1800-1904 (according

to Meinardus). {b) Numbers of ice-character for the Davis Strait region for the period

1820-1930 (according to Speerschneider).

the edge of the East Greenland Current and the North Iceland Current; these ice-
bearing currents often cause the occurrence of severe "ice-years". Ice-rich years recur
rather regularly and there has been a very noticeable ice minimum during the 'forties
and the beginning of the following decade in the nineteenth century. There appears to
be a 4-to 5-year cycle governing the recurrence of ice-rich years ; this is shown quite
clearly in Table 102, which gives mean values based on a 4|-year cycle, beginning
always with a maximum for the period 1880-1904. An accurate determination by
periodograms gave the period of the cycle as 4-76 years. However, the great variability
of the phenomenon does not allow reliable ice-prognoses since the correlation coeffi-
cient of a value with the following fourth and fifth values is only —004 and 0-06
respectively.

Ice in the Sea
Table 102. Severity of ice years around Iceland for a A\-year cycle

269

Period

Before max.
(years)

Max.
0

After max.
(years)

1808-1854
1854-1904

34t
31

24
36t

MX
18

15
14t

35§
44§

16
36

33 1

27
28

Ice-rich
years
(^46)

8§

§ Main maximum; t Secondary maximum; % Minimum

WiESE (1922) found a very high correlation coefficient (r = —0-83 ±0-05) for the
period 1887-1930 between the autumn temperature in north-west Siberia and the ice
volume 4| years later in the East Greenland Current ; lower temperatures are followed
by more ice and vice versa. This relationship is reasonable since A\ years is about the
time required for ice to travel from the Siberian coast to the Greenland Sea. By
comparison of variations in the meteorological elements and secular changes in ice
drift Meinardus was able to show a close relationship with the intensity of the atmos-
pheric circulation. The variations in occurrence of arctic ice in the north-west Atlantic
have been investigated in a series of papers by Mecking (1907, 1939). A series of
observations covering more than 100 years in the Davis Strait have been presented in
the form of "ice character numbers" by Speerschneider (1931) who reduced them on
a ten step scale (Table 103 and Fig. 123 series b).

Table 103. Drift ice in Davis Strait from 1820 to 1930 {ice-
character numbers reduced on a scale from 1 to 10)

Year

1820

1830

1840

1850

1860

1870

1880

1890

1900

1910

1920

Comparison with the values for Iceland shows little similarity. Severe ice years in
one area appear rather to correspond to ice-poor years in the others and vice versa, a
relationship which had previously been pointed out by Schott (1904) correlating the

270

Ice in the Sea

pack-ice occurrence off Newfoundland and that off eastern Greenland. The Davis
Strait values over a series of nine sunspot periods show, however, that the ice amount
in the Davis Strait follows the sunspot cycle with a lag of 2 years rather well (Fig. 124).
The fluctuations in the pack ice in the area of the Newfoundland Banks are of course
directly connected with those in Davis Strait. They also parallel exactly the fluctua-
tions in icebergs in the same area. This is shown by the high correlation factor of

120 5
80 I
40 I

0 s

{

, J

\.\\

'^. ^

(6ii

S
&2

i\f\

vf^

\ UK

\\ y

,'/. \ ';

u /

in'

is'V/

>• *'

'V-

1820

1640

I860 1880

Years

1900

1920

Fig. 124. Relative sunspot numbers and smoothed values for the amount of ice in Davis

Strait. (The latter is displaced two years to the left relative to the sunspot curve (9 full

periods) (full line : sunspot number, dashed line : amount of ice.)

+0-86 between the number of icebergs south of Newfoundland (48° N.) and the pack
ice off Newfoundland valid from February until May (47 years, Smith, 1926-7).

For the Barents Sea particularly good ice statistics are available for areas in which
ice-measurements have been made by the Danish Institute during the years 1896-1916
(Nautik-Meteorol. Aarbog 1916). Wiese (1924) has used these in a study of the rela-
tionship between the occurrence of ice and variations in the atmospheric circulation.
He was able to show that the ice intensity in this sea from May to June depends largely
on the distribution of atmospheric pressure over the Norwegian Sea during the period
from January to the end of April and that a larger (smaller) atmospheric pressure
gradient directed from south-east to north-west between the Norwegian coast and the
axis of the low-pressure trough over the Norwegian Sea causes a decrease (increase)
in the ice coverage of the Barents Sea. By calculations from the regression equations
with the factors affecting the ice coverage, it is possible to obtain reliable ice prognoses
for this area.

A very strong aperiodic change in the Arctic has been in progress since 1918. Since
the summer of that year there has been a general retreat of the ice limit, and at the same
time a warming up of the entire Arctic (Weickmann, 1942). This can be seen best
from the mean position of the ice limit from April to August in the two periods
1898-1922 (25-year mean) and 1929-38 (10-year mean) (Fig. 125). The especially
favourable conditions during the second period are very noticeable when compared
with those for the 25-year mean which can be regarded as normal. Bear Island, for
example, is normally still surrounded by ice in April and partly also in May. During
this second period it was ice-free during all months, and although the northern part
of Novaya Zembla is almost never ice-free the ice limit receded during the second
period almost to the northern tip in July and during August was only a little south
of Franz Josef Land and Wiese Island.

Ice in the Sea

271

20° 40° 60° 80°

80° 80°

70°

65°

75° 75°

70° 70°

65° 65'

Fig. 125. Mean position of the ice limit from April to August in the Barents Sea for the
period 1898-1922 (25-year mean, normal period) and for the period 1929-38 (10-year mean,

warm period).

4. Land Ice in the Sea

(a) Glaciation in Polar Areas

In the polar regions the climatic snow-line Ues so low that under the prevailing
orographic conditions the glacial endings of the ice streams reach the sea and spread
into the ocean. The coverage of polar regions by glaciers was given by Hess and is
shown in Table 104.

Table 104. Glaciation in the polar regions

Area in 1000 kjn^

Northern Hemisphere

Greenland including islands

1896

Spitzberger

Franz Josef Land

Novaya Zembla

Severnoja Zembla

North American islands

100

Total

2131

Southern Hemisphere
Antarctica

13,000

In the Northern Hemisphere the overwhelming part of the total glaciation is on
Greenland where only 0-325 million km^ of its total area of 2-16 million km^ is ice-
free. Glaciers flow out from all sides from the inland ice and a large number of them

272 Ice in the Sea

reach the sea in a broad front. The part played by the other Arctic islands in the
production of icebergs is quite insignificant ; only very few of the ice streams of the
other islands reach the sea as calving glaciers, and even these produce only small
icebergs.

In Greenland the inland ice reaches the sea through more or less narrow fiords
which act as funnels collecting the converging streams of inland ice, but in the Southern
Hemisphere the inland ice reaches the sea in an open front. At the edge of the Ant-
arctic the snow-line is everywhere in or below the sea-level. Here the ice takes the form
of an ice barrier which in the Ross Sea, for example, is about 750 km long and has a
mean height of 36 m, but sometimes exceeds 50 m. Enormous icebergs break away
continually from the edge of the inland ice cover, and though at first often trapped in
the shelf ice, they are carried away with it later on or melt completely in their place.

{b) The Productivity of Glaciers Calving into the Sea in the Arctic

Statistics of iceberg production by glaciers calving into individual oceanic regions
are rather poor; reasonably reliable figures can only be given for very few ice streams.
Smith (1931) has attempted to give such a preliminary survey for the Arctic. In the
Eurasian Arctic there are only a small number of glaciers producing icebergs. In Spitz-
bergen, probably the Negri glacier in the Storfjord; the east coast of North-east Land
has some calving glaciers as has the completely glaciated Franz Josef Land. But the
number of icebergs produced, which are seldom large, is not known and is presumably
small. The productivity of Novaya Zembla and Sevemaya Zembla is equally not
known, but is probably also very small. The few icebergs which are formed at the
islands of the Siberian Shelf move mostly to the west and increase somewhat the
number from the East Greenland icebergs. Smith estimated the number of icebergs
produced annually in the north-east sector of the polar Atlantic ocean as about 600,
which is only about 4% of the annual supply of icebergs from Greenland.

Smith believed that the productivity of the eastern Greenland glaciers was some-
what less than that of the west coast (7500 icebergs per year). There is, however, the
important difference that in the east most of the icebergs are retained in narrow fiords
and are prevented by the solid ice-barrier of the East Greenland Current from drifting
southwards. Their importance to the Atlantic is therefore slight; about twenty to
thirty a year reach Cape Farewell and then drift northwards with the West Greenland
Current. They reach Davis Strait in a collapsing state. The iceberg survey of the
"Marion" Expedition during the summer of 1928 found only seventeen icebergs off
the south-west coast of Greenland; a very small number compared with the enormous
amount that were found in Disko Bay to the north. On the western side of Baffin Bay
only Ellesmere Land with two large ice caps shows any extended inland ice. About
sixty glaciers reach the sea as calving glaciers, but according to Smith the productivity
is not very large (about 1500 icebergs a year). The major source of icebergs is in the
great glaciers of West Greenland from Cape Alexander to Disko Bay. The main part,
from the North-east Bay as far as Disko Bay has more than 100 calving glaciers, the
twelve largest and most productive ones alone producing more than 5400 icebergs
a year. The most important of this group, the Jacobshavener Glacier calves about
1350 icebergs a year into Disko Bay. Not all of these reach the open sea immediately —
on the contrary most are trapped in the fiords for longer periods. In the summer of

Ice in the Sea 273

1928 the "Marion" Expedition found that all the icebergs produced during the pre-
vious 3-4 years (about 4000 to 6000) were accumulated in the Eisfjord. They were all
released from their ice chains during favourable weather at once. Then they arrived
all together in Baffin Bay and drifted slowly to the south.

Table 105. Production of icebergs in different regions on the Arctic

Annual

contribution

North-eastern sector of Atlantic

Islands on the European-Asiatic side

600

East Greenland

7500

From those arriving at Cape Farewell and

passing in the East Greenland Current

20-30

North-western sector of Atlantic

Eastern North America

150

North Greenland

150

Cape Alexander to Cape York

300

Cape York to Svarten Huk

1500

North-east Bay to Disko Bay

5400

Total

7500

Table 105 is a summary given by Smith of iceberg production in the Arctic. This
estimate gives a total annual contribution from Baffin Bay of 7500 icebergs, of which
70% come from North-east and Disko Bays. The table gives only a rough idea of the
ice amount available in Baffin Bay. Direct estimates of the ice outflow from the Green-
land Inland ice by measurement of the speeds of the different glaciers along the western
side of the island still differ widely. De Quervain and Mercanton (1925) estimated
this ice flow as being between 10 km^ and 100 km'' a year. Assuming an average size of a
large iceberg to be about 1-5 miUion m^ and assuming, further, that on calving about
one-third of the ice forms icebergs and two-thirds gives debris and smaller pieces,
then the total mass of ice released on calving is about 4-5 million m^. About 7500
such calvings per year gives approximately 35 km^ of ice. This value lies within the
above Umits. Helland (1876) found values of 5-8 km^ and 2-3 km^ for the annual ice
supply from the Jakobshaven and the Torsukatak Glaciers. Drygalski (1897) found
13-5 km^ for the large Karajak Glacier. According to these figures about 2-6 milhon
m^ of ice are broken off in an average calving of a medium sized glacier ; from this about
one-third is used for production of icebergs.

Careful observation of iceberg calving at the eastern Greenland glaciers led Dry-
galski to distinguish according to the size of the icebergs formed between three types
of calving. The third one proceeds almost continually over several days ; small blocks
of ice break away from the face of the glacier and fall into the sea, often in such large
amounts that the surface is covered with these broken pieces far out into the fiord.
In the second type, large masses are suddenly released in the water from the lower

274

Ice in the Sea

part of the glacier and rise as icebergs to the surface ; this leaves the edge of the glacier
unchanged. The largest icebergs are produced by the first type: under the influence
of the further continuous supply in ice mass the glacier pushes out into the sea for
200-300 m depending on the morphology of the fiord bottom ("fore part of glacier").
The fiord water slowly penetrates into the projecting ice mass and, due to buoyant
forces, the forehead of the glacier gets lifted until it finally breaks off. Calving usually
occurs exactly there where the depth of the fiord has increased to such a rate that the
forward pushing ice-tongue loses contact with the sea bottom and starts floating. In
addition to the increasing buoyancy, lifting due to the tides may also upset the equi-
librium in the glacier tongue. Presumably the formation of icebergs proceeds in the
same way in the Antarctic; however, the process there is of much larger dimension and
produces enormous flat-topped icebergs.

The direct production of icebergs proceeds at about the same rate throughout the
year, but the number of icebergs reaching the open sea depends on the nature of
the fiord and more especially on the season of the year. In winter the fiords are frozen
and the icebergs are trapped. They are released with the coming of summer, all within
a short time and mostly all at once, and they then drift away. This gives rise to the
so-called ""iceberg swarms'" which often occur in Baffin Bay and Davis Strait,

The shape of icebergs is remarkably variable: the pure-chance forms after calving
are remodelled by the action of sea waves and by melting above and below the water;
classification of these diff"erent forms is thus rather pointless. The height of icebergs
varies widely, but the largest are of course found in the area where they are formed.
Measurements made by Drygalski on eighty-seven icebergs frozen into the sea ice in
the East Greenland fiords gave the results shown in Table 106.

Table 106

Height (m)

20-30

30-40 40-50

50-60 60-70

70-80

80-90

90-100

100

Number

6 12

The height decreases rapidly after their formation. The highest iceberg measured by
the International Ice Patrol Service south of Newfoundland was 80 m high; it was
flat-topped and 517 m long. Its volume was estimated as about 25-5 milUon m^.
According to Smith, the icebergs in the Davis Strait have an average volume of 1-5
milUon m^; those of the Newfoundland Banks between 0-1 and 0-15 million m^;
they are about 30 m high. The ''depth of immersion'' of an iceberg depends on the
specific weight of glacier ice. Since icebergs contain a large percentage of air and
numerous cracks and holes this depth does not correspond to that calculated solely
from the specific weight. For mean densities of 0-8997 for the ice and 1-02690 for
polar water, flat-topped icebergs will have one-eighth of their volume above the surface
of the sea and seven-eighths will lie below the surface, but the shape of an iceberg
has a considerable effect on the depth to it which immerses. Smith has made a summary
of direct measurements and has found that for the most peculiarly-shaped icebergs of
of the north-western Atlantic the ratio is 1 : 3, The flat-topped Antarctic icebergs
immerse to greater depths.

Ice in the Sea 275

The destruction of icebergs proceeds by calving, melting and erosion. Icebergs
are often rapidly decreased in size by the breaking away of large and smaller pieces
of ice. This may change the equilibrium of an iceberg so that it capsizes or rolls over.
In cold water the melting process goes on mainly at the water line of icebergs (by
the formation of holes). Melting increases greatly when they drift into warm water
(e.g. south of the Newfoundland Banks in mixed water or in the warm Gulf Stream).
Destruction from above is due to melt water running down the sides of the iceberg,
by erosion and the action of the waves and rain. According to measurements made by
Drygalski in North-east Bay, an iceberg in the summer months may lose from 3 to 4 m
in 7 days. Between Greenland and Newfoundland the ice mass may decrease to an
eighth, corresponding to a daily loss of 1-8 m a day. In the same time the height
decreases by a half.

{d) Iceberg Drift in the Arctic and Antarctic

Icebergs in the open sea are subject to the eroding action of winds and currents.
These effects are dependent: (1) on the ratio of the masses of ice above and below the
water; (2) on the strength and duration of the wind; and (3) on the velocity and direc-
tion of the currents. Mecking (1906) has emphasized the great importance of the wind
and currents for iceberg drift in Baffin Bay. The coastal current plays the decisive part
and the wind determines the course of the icebergs only when this current is weak.
The continuous off-shore wind along the coast of western Greenland in the summer
thereby determines the number of icebergs reaching the Labrador current and thus
the number of icebergs off Newfoundland in the following spring.

The International Ice Patrol Service, in order to determine the influence of the
factors mentioned above on the course of the icebergs, has followed the drift of a
large number in the area of the Newfoundland Banks and has recorded the meteoro-
logical and oceanographic conditions at the same time and Smith (1931) has discussed
this data in detail. The effect of the wind was made up of two parts: (1) the direct
force of the wind exerted on the exposed surface of the iceberg above the water;
and (2) the movement of the floating iceberg with the wind drift set up in the top layer
of the water. For the latter influence it must be kept in mind that for a steady state
the wind drift at the surface of the sea is deflected by 45° to the right of the wind
direction (Northern Hemisphere). This deflection increases with depth and a mean
deflection of 72° can be assumed for the upper 50 m. For the two cases of (a) deep-
immersing larger icebergs and (b) smaller icebergs with immersion ratios of 1 : 1
and 1 : 2 average conditions of the effects of these two forces are given in Table 107
(Fig. 126).

The drift speed of larger, deeper-immersing icebergs with a deflection of 40° to
the right of the wind is less than that of smaller icebergs of lesser depth of immersion
for which the wind force and the force due to wind drift act more closely together.
In this case the deflection from the direction of the wind is only 20°. For more ac-
curate information on the distribution of icebergs in different parts of the sea it is
necessary to make a survey of the existing iceberg accumulations. The International
Ice Patrol Service carried out a systematic investigation of this type with the patrol
boat "Marion" and at the same time the research ship "Godthaab" (Riis Carstensen,

276 Ice in the Sea

Table 107. Direct wind force and force due to wind drift on icebergs

(According to Smith.) fl, deep-immersing large icebergs; b, smaller icebergs).

Direct wind

force in the

wind direction

(km/day)

Wind drift,

deflection 70°

to the right

of wind

(km/day)

Resultant icebergs drift

Wind velocity
Beaufort

Speed
(km/day)

Direction, to

the right of

wind direction

2-6
40

8-8

13-7

3-2
4-8

40
60

4-5
6-9

10-8
16-4

40°
40°

18°
2X'

Fig. 126. Diagram of the forces affecting the drift of icebergs (according to Smith), (a) Effect
of wind on large icebergs; (b) Effect of wind on small icebergs.

1929, 1936) made an oceanographic survey in Baffin Bay. Figure 127 shows the distribu-
tion of icebergs in the Davis Strait and the Labrador Sea during the summer of that
year. The few icebergs along the south-west coast of Greenland are from the East
Greenland Current. Most of the icebergs are carried southwards by the cold Labrador
Current which runs close to the coast. The central parts of Davis Strait and the Labra-
dor Sea are almost completely free of icebergs. The Labrador Current along the coast
thus forms the channel along which the icebergs pass towards Newfoundland. The
track of the icebergs, especially to the east and south of Newfoundland, has been

Ice in the Sea

277

accurately fixed by tracking numerous icebergs with the patrol ships. The main ice-
berg track as shown by these detailed surveys is shown in Fig. 128. An increased fre-
quency is to be expected along the eastern slope of the Newfoundland Banks where the
Labrador Current turns towards the west and its cold and weakly sahne water mixes
along the southern side of the current in large eddies with the warm and highly saline
water of the Gulf Stream. A careful study of these eddies by the Ice Patrol vessels has

N 60

Fig. 127. Extent and distribution of icebergs in Davis Strait and the Labrador-Sea in the
summer of 1928 according to the "Marion" Survey.

been made and thereby an explanation was found for the continuing presence of ice-
bergs in this part of the sea, since the eddies keep the icebergs quasi-stationary
Occasionally individual icebergs withstand the destructive effects of the warm At-
lantic water and reach much further south than usual. The most southerly position
so far recorded was 30° 20' N. and 62° 32' W. near the Bermudas for an iceberg about
9 m long, 5 m broad and 1 m above the water, which was sighted by the "Baxter-
gate " on 5 June, 1926.

Knowledge of iceberg drift in the polar seas of the Southern Hemisphere is very
scanty. The approximate northern limit of drifting icebergs is shown in Fig. 122. It
is, of course, far north of the northern limit of drifting ice floes since the compact
mass of a large iceberg can better withstand the destructive action of warm water and
air. It must be assumed that here also winds and currents must be the factors that
determine the drift of an iceberg. In some individual cases a relationship to the course
of low-pressure areas has been demonstrated, but in view of the irregularities of the
latter a strict relationship is hardly to be expected (Mecking, 1932).

Icebergs are especially important in the Falklands area where they are sometimes
carried, accompanied by drift ice, far to the north in large numbers. They have been
sighted as far north as 42° S. and in 1906 even reached as far as 37° S. (59° W.). The
aperiodic variations in the occurrence of ice appear to be particularly large here. In

278

Ice in the Sea

Fig. 128. Main iceberg tracks off Newfoundland and the Grand Banks.

the years 1891, 1892, 1893 and 1906 a remarkable accumulation of icebergs appeared
in the area south of Cape Horn and northward of the Falkland Islands as far as 40° S.
They occurred mainly along the edge of the shelf; farther to the west they were com-
pletely absent. They are trapped and melt rapidly inside the numerous eddies along
the boundary between the Falklands Current and the Brazil Current in a similar way
to those south of the Newfoundland Banks.

(e) Seasonal and Aperiodic Variations in Iceberg Frequency off Newfoundland

Surveillance of the distribution of icebergs in the area of the Newfoundland Banks
since 1900 has given the mean annual iceberg frequency and its variation from month
to month shown by the data in Table 108 for Newfoundland (south of 48° N.) and
for the area south of the Grand Banks.

Table 108. Mean annual variation in iceberg frequency (a) off Newfoundland south of

48° N, and (b) south of the Grand Banks

(For the period from 1900-26)

Month

Jan. Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Total

(a)
ib)

3
0

83
9

130

68
13

25
3

4
0

3
0

2
0

386
51

Ice in the Sea

279

The iceberg season usually lasts from 15 March to 15 July, but the number of ice-
bergs decreases rapidly after the middle of June (see Fig. 117). From the middle of
July to the following spring the area south of the Grand Banks is almost free of ice-
bergs. The variations in iceberg frequency from year to year are very large. South
of 48° N. there were in 1929 a total of 1351 icebergs, in 1924 only eleven. An accurate
monthly record of these values is available starting from 1900. Together with the
previous data compiled by Mecking there is now a complete series of records available,
covering a period of 50 years for the iceberg frequency off Newfoundland. This is
shown graphically in Fig. 129. With these more or less homogeneous data it is possible

- i 1

I \

k 1 - ■

\aM^

rl'vik il

^t 1 H<

'\\

/ ''

i '

l/*i

'/ \

}

1880

1890

1900 1910

Years

1920

1930

Fig. 129. Character of the iceberg-years from 1880 to 1930 off Newfoundland (ten step-
scale, according to Smith).

to investigate with some hope for success the causes of the aperiodic variations in
numbers of icebergs. In the first place there appears to be a correlation between the
atmospheric pressure gradient from Iceland to Greenland-North America a few
months previously and the iceberg maximum off Newfoundland. Determination of
the air pressure anomaly for the North Atlantic for the months December to March
during 6 years of low iceberg frequency, in this case with a total of 275 icebergs, and
for the same months in 5 years with a high iceberg frequency with a total of 774
showed completely opposite conditions (Fig. 130). A weak Icelandic low pressure area
during the spring and the autumn with a weak pressure gradient over Baffin Bay and
Davis Strait seems to be followed by the low-frequency iceberg years. During ice-
berg-rich years, on the other hand, the Icelandic low-pressure area is intensified and
the strong pressure gradient to the west is accompanied by strong air movements and
stronger wind drift in the iceberg area along the North American coast. Smith has
also tried a quantitative determination of this relationship using the correlation
method, and has obtained a prognostically valuable formula. An increase in the ice-
berg frequency in the north-western Atlantic is thus accompanied by an intensification
of the atmospheric circulation in the polar areas which corresponds to an increase in
the outflow of polar air and of the arctic water towards the south. These relation-
ships of course take into account only the meteorological effects and not the possible
fluctuations in the production of icebergs by the western Greenland glaciers. At the
present time it is not possible to make an estimate of these.

5. EflFect of Polar-Ice Conditions on the Atmospheric and Oceanic Circulation

The total annual ice outflow along the whole of the west coast of Greenland has
been estimated by Smith as between 42 and 63 km^ (see p. 272) ; the American coast

280

Ice in the Sea

80° 60° 40° 20° 0° 20° 40° 60°

80°

100° 80° 60° 40° 20° 0° 20° 40° 60°

100°

Fig. 130. Iceberg frequency off Newfoundland and atmospheric pressure anomaly over the

North Atlantic.

Ice in the Sea 281

adds only about 1-9 km^. This is the amount of land ice that exists in Baffin Bay on a
yearly average and drifts southward to melt in Davis Strait, along the Labrador coast
and in the Newfoundland area. The amount of sea ice melting during one year can be
calculated from the average area covered by pack ice and drift ice. Smith has made an
estimate of this kind based on reliable data collected by the Ice Patrol cruises. The bases
of this are contained in Fig. 131 which also shows the areas which stand in question;
the most important are the shelf areas where the ice-covered area is about 1-6 million
km^. Taking the mean thickness of drift and pack ice as about 1-8 m, the total amount
of sea ice will be about 3000 km^. In contrast to this, the land ice amounts to only
44-65 km^, so that of the average annual amount of ice melting in the north-west
Atlantic only between a hundredth and a two-hundredth part comes from icebergs.
This is vanishingly small (see Fig. 131). This comparison shows that the amount of
pack ice and drift ice is the decisive factor. If for any oceanographic or meteorological
problem a consideration of the effects of ice destruction in the north-west Atlantic —
which vary considerably from year to year — is needed, it is thus not justifiable to
compare it with variations in the ice frequency, as has often erroneously been done.

In dealing previously with convection processes (see p. 97) two possibilities were
discussed for the initiation of such a process, which are of the greatest importance to
the thermal structure of the middle and bottom layers of the oceans. It was assumed
by Pettersson that the necessary heat loss of the upper water layers was mainly due
to the melting of ice in polar and subpolar oceanic regions. However, laboratory
experiments by Nansen showed that this hypothesis was untenable. For the special
case of the conditions in the north-west Atlantic it is possible, using the values given
by Smith to determine directly the amount of heat which is required for the observed
yearly melting of pack ice and drift ice and therefore is not available for heating the
ocean and the atmosphere. This can be compared, as has been done by Smith, with
the amount of heat suppHed during the summer by solar and sky radiation which is
required for the increase in temperature of the upper 150 m layer of water (the
average depth to which the increase reaches downwards into the sea). From the num-
bers given in Fig. 131 it can be seen that the mean summer increase in the tempera-
ture of the water masses in this area (down to 150 m) is about 1-2°C. It can also be
calculated that the annual melting of pack ice and drift ice in the same area is sufficient
to decrease the temperature of the layer down to 150 m depth by 0-6 °C. Thus in the
north-western part of the North Atlantic the water is cooled by the melting of the
ice by only about half of the amount of the summer increase in temperature due to
the absorption of solar and sky radiation. Dynamic treatment of the oceanographic
data of the "Marion" and "Godthaab" Expeditions permits the calculation of the
amount of the heat deficit at the Newfoundland Banks due to the continuous supply
of cold polar water by the Labrador Current. Comparison of this heat reduction with
that due to ice melting shows that the latter accounts for only 10% of the cooling
effect of the Labrador Current. The dominant factor in the cooling of the water masses
of the northern part of the North Atlantic is thus neither the mehing of icebergs nor
of the pack ice and drift ice, but much more the continuous advective supply of
polar water which the Labrador Current carries southwards towards the warm water
of the Gulf Stream.

The "Meteor" cruise in Icelandic and Greenland waters have given the same

282

Ice in the Sea

Fig. 131. a quantitative representation of a number of comparisons between ice-melting
effects and related phenomena. The shaded area bounded by the full line in the normal pack-
ice area. The dotted line marks the mixing zone. The entire melting area, with a uniform
thickness of 150 metres is divided into six parts; in summer the southernmost is heated an
average of 5°F, and the northernmost only 0-5''F. The spot "M" off Cape Farewell repre-
sents the annual crop of glacial ice expressed in the same scale as the pack ice and as one
large berg. The shaded area 'W represents the total annual discharge of glacial ice into
BaflSn Bay, expressed on the same scale and in terms of pack ice 6 ft thick.

Ice in the Sea 283

results (Defant, 1933). The formation of the East Greenland Current and the main-
tenance of its polar character as far as Cape Farewell is not due to melting processes ;
its Arctic nature is mainly acquired from its direct connection with the North Polar
Basin causing a continuous supply of polar water and from the climatic conditions
maintained over Greenland by the inland ice. This advective supply of Arctic water
from areas where the effect of solar radiation is very small is the determining factor,
and sea ice and icebergs are only accessory phenomena.

A marked effect of the ice masses of the polar seas on the atmospheric circulation
has been assumed by many prominent meteorologists. Hildebrandsson (1914)
especially has attempted to show that the cause of the secular variations in meteoro-
logical factors is to be seen in the aperiodic variations in the amount of polar ice.
More recent data from later investigations has lent support to this hypothesis, but a
definite proof is difficult. Both phenomena are not independent of each other, so that
it is reasonable to assume, of course, a mutual interaction between the ice conditions
and the atmospheric circulation; it is not easy to separate cause and effect (Wiese,
1924). Conditions are probably such that variations in the atmospheric circulation
change the equilibrium conditions in the polar reservoirs of cold air. Years with
weaker circulation favour an increase in the thickness of the cold air masses in the
polar regions. This increases the atmospheric pressure in the polar region and corre-
spondingly winds and currents become stronger, which causes a greater extension of
the polar ice towards the south. The increased ice surface in turn increases the air
pressure; the atmospheric pressure anomaly thus acquires a certain permanence, and
due to this mutual reinforcement the effect may last a long time. The atmospheric
pressure in the polar areas is thus a very sensitive indicator of the general condition
of the atmosphere. Since, however, the atmospheric pressure conditions in these re-
gions is reflected in the ice conditions, the distribution of ice in the polar seas can be
taken as a measure of the variations in the general atmospheric circulation, provided
sufficiently accurate information is available.

The major variations in the atmospheric circulation usually extend throughout
the entire atmosphere over the whole Earth, both in the Northern and Southern

Table 109. Parallelism between changes in ice conditions of the north

and south polar regions
Shown by the relation between ice conditions from March to May at the South
Orkney Isles (years with close or open ice) and corresponding deviations of the
ice coverage in May to August from an average value of the period 1896-1916

in the Barents Sea

South-Orkney-Isles

Character of ice

conditions for March

to May

Close ice

{

Open ice . . . .■!

Barents Sea
deviations of the ice coverage (in 1000 km^) for May to
August from an average value of the period 1896-1916

1903 1909 1910 1911 1912

+88 +102 +17 +97 +176 (above average)

1904 1905 1906 1907 1908
-158 -165 -61 -130 -121 (below average)

284 Ice in the Sea

Hemisphere. Thus, for example, it has been shown with sufficient certainty that there
is a high positive correlation between the atmospheric pressure pulsations in the North
Polar regions and those in the South Polar regions. If this connection is real, certain
parallelism would be expected between the variations in ice conditions in the Arctic
and in the Antarctic. To test this assumption Wiese has compared 10-year records of
ice conditions at the South Orkney Islands from 1903 to 1912 (Mossman, 1923)
between March and May, with the area of ice in the Barents Sea between May and
August in the same years. The results are given in Table 109 and show the existence
of a positive correlation.

It is obvious that such a relationship between Arctic and Antarctic ice conditions
can only be investigated by means of the indirect method of an investigation of
variations in the general circulation of the entire atmosphere; it shows, however, the
great scientific and practical importance of a continuous systematic observational
check on ice conditions in the polar regions.

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Part 2
DYNAMICAL OCEANOGRAPHY

Introduction

Dynamical oceanography is concerned with the movements of the water masses of
the oceans. In addition to the framework of the vertical and horizontal structure, in
the sea, of the oceanographic factors such as temperature, salinity and density dealt
with in detail in Vol, I, Pt. I, we have now to consider the forces present that cause
displacements of the water masses. These displacements are termed ocean currents;
they are phenom.ena that an observer will be directly aware of only occasionally,
near land or in narrow straits. In the open sea they are shown only by calculations
carried out for quite different purposes and only give a clear picture of the movement
of the water masses when taken together over a larger area of the sea. The system of
currents in the ocean, hke that in the atmosphere around the Earth, is among the most
striking phenomena in geophysics. The oceanic circulation involves the whole ocean
and the conditions are aptly described by Heraclitus' expression -navTa pTt (it all
moves).

In principle, dynamical oceanography can be subdivided into two main parts.
One concerns ocean currents in the more restricted sense of the word as applied to the
steady continuous transport of water in definite direction. In these currents movements
in a horizontal plane predominate overwhelmingly, but there are also phenomena
where a vertical component becomes important. The second part of dynamical
oceanography concerns the phenomena associated with periodic water movements in
which the whole process is repeated after a certain time. These are the waves and
tides. Separate treatment of ocean currents (in the more restricted sense) on the one
hand and of the waves and tides on the other considerably simplifies their presentation,
although these phenomena are not separated in nature to the extent that might super-
ficially be easily assumed.

Part II of this volume is therefore devoted mainly to ocean currents (in the narrower
sense) while Volume II deals with the dynamics of the periodic phenomena (Waves
and Tides).

An explanation of the movements of water masses in the ocean requires in the first
place a study of the interplay between the oceanographic factors and of the effect of
external forces on the water masses. It is self evident that hydrodynamics must play a
major role in dealing with these questions, especially if the problems arising are treated
more from a geophysical standpoint. Thus, in addition to a more statistical-geographi-
cal description of observed oceanic phenomena, hydrodynamic considerations have
to be used and finally one attempts to explain them on a physical-mathematical basis.

Ordinary classical hydrodynamics develops the theory of movement in a liquid on
the assumption that it is homogeneous and incompressible. As a first approximation
the results of pure hydrodynamics are applicable within wide limits to the water

299

300 Introduction

movements in the sea and allow a deeper insight into the basic causes of the phe-
nomena observed. This is not entirely sufficient, however, because water masses of
the ocean show quite large deviations from ideal homogeneity and their density de-
pends strongly on the temperature and salinity. It is therefore necessary to take into
account the stratification and compressibility of the water masses. Classical hydro-
dynamics is thus replaced by physical hydrodynamics, for which at the present time
only for some of the more important parts a precise theoretical basis has been de-
veloped. (Bjerknes and co-workers, 1932 or Godske, Bergeron, Bjerknes and

BUNDGAARD, 1957).

Chapter IX

The Geophysical Structure of the Sea

1. Introduction

Since the ocean currents are displacements of water masses the distribution of mass
in the sea becomes of particular importance in all hydrodynamic investigations. It
is specified by the distribution of the density or of the specific volume, which are both
determined by the thermo-haline structure of the ocean. In addition to the distribution
of mass there is an internal field of force due to the distribution of pressure of the
water masses in both vertical and horizontal direction. The atmospheric pressure which
exerts a varying force on the surface of the sea must be also considered a source for
disturbances. In addition to these forces the only external conservative force, the
gravity, must also be taken into account, since it intervenes in an essential way in all
phenomena involving the movement and equilibrium of the water masses of the oceans.
Thus the fields of gravity, of pressure and ofrjiass in the ocean play an important part
in all hydrodynamic investigations. For a quantitative description of a phenomenon
the magnitude of the units used is essential. At the present time the absolute units of
the CGS system [cm, g, sec] are preferably used, but in many cases according to the
magnitude of the numerical values there are practical advantages in the use of larger
units, usually obtained by multiplication by a suitable power of 10. The metre (10^ cm)
is a suitable unit of length in dynamic oceanography, but the nautical mile, which is
the length of an arc of one minute at the equator (1852 m), is also used. The metric
ton, the mass of a cubic metre of water with a density of 1 (10® g) is frequently used as
mass unit together with the second, the minute and the hour as units of time. The
velocity may be defined in absolute units (cm sec~^) but is also frequently given in
nautical miles per hour (= 1 knot = 51-4 cm sec~^) (Maurer, 1938).

2. The Distribution of Gravity and Gravity Potential

Gravity is the result of the force of attraction of earthly masses and of the centri-
fugal force of Earth rotation. Its distribution at the surface of the Earth can be found
by pendulum measurements, for which it is suflftcient to use its normal values. At the
present time the most frequently used formula for a calculation of gravity is that of
Heiskanen and Cassini; Lambert, 1931 :

g^ = 978-049 (1 -f 0-0052884 sin^ cf, - 0-0000059 sin^ 2<^ [cm sec-^]. (IX. 1)

Calculations of the gravitational acceleration within the sea must take into account the
density of the water mass and also the result of the potential theory that the outer
shell of a sphere exerts no attraction on a point in the interior of the sphere. If k is
the gravitational constant, M the mass of the Earth and R the radius of the Earth

301

302 The Geophysical Structure of the Sea

then, as a first approximation, the gravitational acceleration at sea-level is given by

while at a certain depth h where m is the mass of the Earth shell {R — h) it reduces
to

M — m

* {R - hf '

so that in a first approximation

^0 + 2gQ -

m R

^ ~ 2M h

If the mean density of the Earth p,„ is 5-5 and p is the density of the shell
{M = 3(47T^R^)pm and m = A-rrR-hp], then the expression in brackets becomes

-^ , P ~ 1-05 and ^ = 3-086 x 10-^.

^ Pm R

In that way one obtains for the change in gravity within the sea the relation

g = g^^ 2-303 X 10-«r, (IX.2)

where z is the depth of the point under consideration.

Equations IX. 1 and 2 give the intensity of gravity, and its direction is fixed by the
direction of the plumb-line. A surface which is always at right angles to the direction
of gravity is termed a level surface (Niveau-Flache). Considering only the gravita-
tional force, no work will be expended in the displacement of a body along such a
surface. The most important gravitational level surface is the free surface of the sea,
the ideal sea-level (see p. 6), wliich forms a part of the geoid. Every other level
surface is uniquely fixed by the amount of work that must be expended in moving a
particle from the ideal sea-level to any point on the surface under consideration. For
a surface at a depth h this work measured along the plumb-line direction is given by
the product gh. The level surfaces are thus also surfaces oi equal gravitational potential
with the ideal sea-level as principal potential surface with zero potential. In this way
the entire oceanic space can be regarded as intersected by a finite number of equi-
potential surfaces each of which is separated from the next one by a unit potential
layer. The thickness of this layer varies as g alters from point to point but the product
gh must always remain constant. The level surfaces must carefully be distinguished
from surfaces of equal depth below the sea surface. The two sets of surfaces will inter-
sect, and where a surface of equal depth is not at right angles to the vertical there will
be a gravitational component in the direction of this surface. If the two surfaces
were solid and smooth, a ball on a level surface would remain at rest, but on a sur-
face of equal depth it would begin to roll away from the equator towards the pole
under the influence of the gravitational component directed towards the poles.

A point within the sea may be fixed by taking three co-ordinates, either (1) ^ the
geographical latitude, A the geographical longitude of the projection of the point
under consideration on the surface of the sea along the plumb-line and //, the geo-
metric depth of the point itself or, (2) the co-ordinates (j) and A as in (1) and as third

The Geophysical Structure of the Sea 303

co-ordinate the potential value gh at the point under consideration (as a positive
quantity). In the first case, all points v^^ith the same third co-ordinate will lie on sur-
faces of equal geometric depth and in the second case the points of equal third co-
ordinate gh will lie on a level surface. The second system of co-ordinates is much
more suitable for problems of the statics and dynamics of the ocean, since at every
point in such a co-ordinate system the total force of gravity acts in the third direction ;
there is no component of gravity acting along the other two. As g is approximately
10 m sec-2 the potential gh will change by one unit if the unit mass is lowered by
about 1/10 m. that means when the depth is reduced about by 1/10 m. Bjerknes
(1910, 1912) has denoted this unit potential the dynamic decimeter (1 dyn.dm).
Multiples and fractions of it are the dynamical metre ( 1 dyn .m) or the dynamical centimetre
(1 dyn. cm), respectively. By the introduction of this potential quantity as the third
co-ordinate the level surfaces become surfaces of equal dynamic depth.

The dynamic depth has the dimensions [g cm^ sec^^]. The most practical unit of
the dynamic depth is the dynamic metre. If /; is expressed in metres then the dynamic
depth D in dynamic metres is

Z) = f^, (IX.3)

and at this point there is a geopotential

0 = -10Z>. (IX.4)

Since the gravitational acceleration g changes with depth according to (IX. 2) the
difference between two dynamic depths in the ocean is given by the relation

g dh. (IX.5)

10
As a first approximation (IX. 3) thus gives

D = 0-98/2 and /; = 1-02Z). (IX.6)

The numerical difference between a dynamic metre and a geometrical metre is thus
about 2%. Tables for converting one unit into the other according to more accurate
formulae have been given by Bjerknes and co-workers (1912, 1913).

3. The Field of Mass

The mass field is given by the distribution of the density p or its reciprocal, the spe-
cific volume a. In the sea it can be represented in a suitable way by surfaces of equal
density {isopycnic surfaces) or by surfaces of equal specific volume (isosteric surfaces).
The latter are used preferably in oceanography. The field of specific volume a^^, „ can
be regarded as made up of two separate fields. The first of these agg, o, p represents the
mass field of a homogeneous sea at 0°C and 35%o S (standard ocean) ; it is in this way
completely defined and invariable. The second is the field of the specific volume
anomaly S and this set of surfaces of equal anomaly 6 is quite sufficient for the charac-
terization of the mass field in the total oceanic space.

In a vertical section of the mass distribution the isosteres and the isopycnals appear
as curved or wave-form lines deviating only slightly from the horizontal. A large

304 The Geophysical Structure of the Sea

exaggeration of the vertical scale is required to show the slope of the lines in a better
way. The geometrical depth, the dynamic depth or the pressure can all be used as the
vertical co-ordinate. Such graphic representations are termed dynamical vertical cross-
sections, in short dynamic sections.

4. The Pressure Field and its Relationship to the Mass Field. Solenoids

The internal stress in a liquid such as the ocean is characterized by the pressure per
unit area. In a liquid in equilibrium, due to the absence of any resistance to deforma-
tion, this pressure acts perpendicular to any arbitrarily oriented surface through the
liquid and is equal for any point and in all directions. This state is denoted as hydro-
static stress state. The water masses in an ocean at rest is subject to the influence of
gravity and the static pressure p at a depth h is defined as that force produced by the
weight of a water column of unit cross-section extending from this depth to the surface
of the sea. This does not take into account the atmospheric pressure at the surface
of the water so that p is defined solely as the water pressure. Thus

P = pmgh,

where Pm is the mean density of the water column /;. The dimensions of p is
[g cm^^sec"^]. According to (IX. 3) the dynamic depth D can be substituted in place
of the geometric depth /; so that

P = PmD. (IX.7)

The pressure of a column of pure water (p„j = 1) of a height of 1 dyn. m is defined
as 1 decibar. This is a tenth part of a bar which is defined as 10^ dyn/cm^ and is the
pressure of a column of pure water of lOdyn.m. The practical pressure unit "one
atmosphere", is only about 1% greater than one bar. Fractions of the bar in addition
to the decibar are the centibar and the millibar. The latter corresponds to a water
pressure of one dynamical cm of pure water and is equivalent to a pressure of 0-75
mm of mercury.

For an ocean of pure incompressible water the following rule applies: The numerical
value of "sea pressure" expressed in decibars is the same as that of the depth in dy-
namic metres at which this pressure is exerted. Since p^ in the sea is not very diff'erent
from 1 this rule also applies in very close approximation for sea-water. From equation
(IX. 7) is obtained

D = a^p, (IX.8)

where a,„ is the mean specific volume of the water column. If p or a vary, equations
(IX. 7 and 8) will be replaced by the integral forms

\ pdD and Z) = a

p=\pdD and D=\adp, (IX.9)

where the integrals must be extended over the whole water column h. For numerical
calculations the integral is split up into sums for the thinnest possible layers with
approximately constant density or specific volume (see later).

The relationships between pressure, geometrical and dynamic depth and the vertical
distribution of specific volume and of density are shown in Table 110 for a homo-
geneous sea at 0°C and 35%o salinity (standard ocean).

The Geophysical Structure of the Sea

Table 110, Vertical stratification of a homogeneous ocean at 0°C and 35%o
salinity (standard ocean)

305

Pressure

Geom.

Dynamic

Spec.

Density

Dynamic

Pressure

depth

volume

depth

(dbar)

(m)

(dyn.m)

(lO^a)

(<Tt)

(dyn.m)

(dbar)

97264

28-23

100

99-24

97-242

97219

28-61

100

102-837

200

198-45

194-438

97174

29-12

1 200

205-724

300

297-60

291-590

97129

29-64

i 300

308-659

400

396-71

388-696

97084

3003

400

411-643

500

495-78

485-758

97040

30-50

i 500

514-677

600

594-80

582-776

96995

31-02

600

617-758

700

693-77

679-749

96951

31-45

700

720-889

800

792-69

776-678

96901

31-92

1 800

824068

900

890-57

873-564

96863

32-41

1 900

927-296

1000

984-41

970-404

96819

32-85

; 1000

1030-572

1500

1482-97

1453-955

96602

35-17

1500

1547-696

2000

1975-43

1936-429

96388

37-47

2000

2065-967

2500

2465-96

2417-836

96177

39-75

2500

2585-445

3000

2956-20

2898-204

95970

41-99

3000

3106-094

3500

3445-55

3377-544

95766

44-21

3500

3627-903

4000

3932-89

3855-873

95566

46-40

4000

4150-862

5000

4904-57

4809-556

95173

50-72

5000

5200-185

6000

5873-38

5759-368

94791

54-95

6000

6253-981

7000

6836-43

6705-421

94421

5908

' 7000

7312-174

8000

7796-89

7647-817

94060

63-15

j 8000

8374-688

The difference between the numerical values of the pressure in decibars and the
geometrical depth in metres is of the order of 1 % and remains the same also for other
thermo-haline vertical stratifications. It is thus permissible, as a first approximation, to
ignore this difference: at a depth of « geometrical metres there will be a pressure of
n decibars. On the other hand, the difference between dynamic and geometric depth
in metres is about 2%, and between dynamic metres and the pressures in decibars is
almost 3%. For hydrographic purposes these differences are too large to be ignored.
For rough calculations it is perhaps permissible and practical to approximate the
values of Table 110 by the following formulae:

lO^a = 97264 - 0-44/7,

lOV = 102823 - 0-46/7.

The values calculated from these are accurate to some units in the fifth decimal place.
The pressure field can be represented by surfaces of equal pressure (isobaric surfaces).
If these are drawn for each decibar then the entire volume of the sea is divided into
layers of 1 decibar pressure difference. The pressure gradient G at any point on an
isobaric surface is given by the decrease in the pressure p along the normal n to this
surface

G = -^

(IX. 10)

306 The Geophysical Structure of the Sea

In general, the isobaric surfaces and the surfaces of equal dynamic depth (level sur-
faces) intersect. These lines of intersection are termed dynamic isobaths and are
usually plotted at 5 dyn.mm intervals. In this way the topography of the pressure
surface is obtained. On the other hand, the lines of intersection of the pressure surfaces
with a level surface are denoted as isobars or lines of equal pressure. These give a
chart of the pressure distribution at a given level. In oceanography it is more cus-
tomary to represent the pressure field by charts of the dynamic topography of espe-
cially selected isobaric surfaces.

It should be emphasized that for a representation of the pressure distribution in the
ocean only the actual water or sea pressure is used without taking the air pressure into
account. If the total pressure is required the sea-level pressure of the atmosphere
which on a crude average is about 10 decibars must be added. Furthermore, it should
also be remembered that dynamic topographies are referred to the physical sea-level
from which the measurements are made and not to the ideal sea level (the geoid)
which is defined as the surface of zero gravitational potential (dynamic depth zero).
The topography of the physical sea-level is unknown, so that in practice these
topographies are always represented only as relative topographies, i.e., relative to the
unknown topography of the physical sea-level. Expressed in another way, they
are dynamic topographies relative to a physical sea-level assumed as "plane" (plane
in a geodetic sense). In order to obtain the absolute dynamic topography, the
absolute dynamic topography of the physical sea-level would have to be known,
and for this a determination of the dynamic depth of the pressure values would have
to be carried out starting from the physical sea-level.

A convenient and practical representation of the mass distribution is obtained by
use of dynamic sections — or to be more specific, vertical sections — of pressure sur-
faces and the isosteric surfaces. Both of these sets of surfaces vary only slightly from
the horizontal, and the vertical scale must be considerably exaggerated in order to
obtain visible gradients. Usually, however, the inclination of the isobaric curves as
compared with that of the isosteric ones is so slight that horizontal lines in the co-
ordinate system can be taken as isobars. The specific volume anomaly is usually
used instead of the specific volume itself and the mass field is therefore represented by
lines of equal anomaly.

The two sets of curves (the isobars and the isosteres) divide the vertical surface into
a number of parallelograms formed by wavy lines; they are the cross-sections of tubes
formed by the intersection of (invariably) two isobaric surfaces and two isosteric
surfaces. These differently-shaped parallelepipeds were denoted isobaric-isosteric
tubes by Bjerknes (1900); they are denoted as unit tubes or solenoids if areas of units
in pressure and specific volume are drawn on vertical sections.

The terminology "solenoid" is also used when the sets of curves are drawn at inter-
vals of several units. If the mass field is given by Hnes of equal anomaly S at unit inter-
vals of a (in the CGS system: 10~^), and the pressure field by isobars at intervals of
1 db (in the CGS system: 10^ dyn. cm^^), then a parallelogram formed by intersection
of two isosteres and two isobars will enclose one solenoid of the CGS system. In
practice, isosteres are usually drawn for every 20 of these units so that a surface ele-
ment of the isobaric-isosteric tube contains 400 CGS solenoids. The solenoid is
assigned a positive or negative sign depending on whether, on rotation in a positive

The Geophysical Structure of the Sea

307

sense (anticlockwise) on the isostere with the higher value for the specific volume, the
higher pressure comes before or after the lower. The solenoids have the same proper-
ties as the isobaric-isosteric tubes ; they must be either fully enclosed or must terminate
against a boundary surface. In the case of hydrostatic equilibrium the two sets of
surfaces will not intersect and there will thus be no solenoids. On the other hand, as
the incUnation of the two sets of surfaces relative to each other increases, the number
of solenoids will also increase, so that their number can be taken as a measure of the
deviation of this state from hydrostatic equihbrium. Since the isobars in practice
appear in the dynamic section as horizontal lines, the number of solenoids in a section
enclosed by a closed curve is determined primarily by the degree of concentration of
the isosteres and their slope. The number A'^ of solenoids within a closed curve s is
given by the equation

(IX.ll)

A^ =

a dp.

where the integral is taken along the curve 5 in a positive sense of rotation. This is
easily understood if the oblique-angled co-ordinate system of the p- and a-lines is
transformed into rectangular co-ordinates, with the /?-values as abscissa and the a-
values as ordinate.

Of particular interest is the case of a curve s formed by two vertical lines and two
isobars. The first two correspond to lump-lines at two oceanographic stations a
and b, the latter two represent the intersection of the two dynamic topographies of
certain pressure surfaces. The pressure at the upper isobaric line at sea-level will be
/?o, the pressure at the lower one p^, and will occur at station a at the dynamic depth
Da, and at station b at the dynamic depth D^ (see Fig. 132). Since along the two iso-

FiG. 132. To the computation of the number of solenoids enclosed by the curve aa' bb'.

baric lines dp = 0 these two parts of the curve s will not contribute to the integral
in (IX. 1 1), so that

— (h a dp

a dp

However, from the definition of equation (IX. 9)

a dp

and Dt

a dp

308 The Geophysical Structure of the Sea

so that (IX. 12) becomes

N = Da- D„

i.e. the difference in the dynamic depth of an isobaric surface at two oceanographic
stations gives the number of solenoids in the cross-section between the two stations
from the surface of the sea to the depth of the isobaric surface.

If the sets of surfaces of two properties of sea water L^ and Lo coincide, there must be
a functional relationship F(Li, L,) = 0 between them ; this represents only a purely
geometrical connection between the two scalar quantities Li and Lg and reveals nothing
of the physical relationship which probably exists between them. Examples of such
quantities are p, a, p and also the potential 0 or the dynamic depth D. All these sets
of surfaces coincide only when there is internal equilibrium in the water mass (see
p. 302). The field of the second scalar quantity Lg was denoted by Bjerknes and co-
workers (1933) in the case where L^ = p == const, (isobaric surfaces) barotropic,
that means adjusted to the pressure field ; in the case where Li= t = const, (isothermal
surfaces) thermotropic, where it is adjusted to the temperature field and in the case
where L^ = S = const, (isohaline surfaces) halotropic, where it is adjusted to the
salinity. There will then be a set of relationships

F(p, L,) = 0, F{t, L,) = 0, F(S, L,) = 0.

In general, these relationships are denoted "conditions of homotropy". If no such
functional relationships exist, then the field of the scalar quantity Lg is barocHnic,
thermoclinic or haloclinic, i.e. it is "inclined" relative to the pressure field, the tempera-
ture field, or the salinity field; only in these cases do solenoids exist.

If jc is a definite point in a field and x + dx is a neighbouring point, and if the geo-
metrical changes on transition from one point to the other are AN^^ Ap, At, AS then
the quantities

J^~~A^ ~ " dFfm, ' ^^~ At " dFjdN^ ' ^^~ AS ~ dFjdN^

are termed homotropic coefficients of N2, and specifically each as the barotropic,
thermotropic and halotropic coefficients. These coefficients are entirely geometric in
character, since they depend on the differences of the factors at two diff'erent spatial
points. The behaviour of an individual small particle — e.g. on changes in pressure —
is on the other hand a purely physical property of the field; for example, the density is
given by the piezotropic coefficient of density

dp 1 da

'^"^dp^^^dp'

the changes in p and a on displacement of a small particle depending on the change in
pressure dp. The diff'erence between the two coefficients is clearly shown by the follow-
ing example: Fp^ = (ApjAp) = 0 indicates homogeneity of the mass field, while
yp = 0 indicates the incompressibility of the medium. Bjerknes termed the special
case Tp" = yp "autobarotropy", i.e., after exchange of any two small particles the mass
field remains unaltered.

The Geophysical Structure of the Sea 309

5. The Dynamical Method of Preparation of Oceanographic Data

The oceanographic measurements made at a station give the thermo-haline structure
of the sea at this place in terms of temperature and salinity at definite depths. The
dynamic evaluation of this data includes the determination of the density and the
specific volume in situ at definite standard depths, and, in addition, the calculation
of the pressure for given depths and the dynamic depths at given pressures, respectively.
The starting point for this is the integrals of the two equations (IX. 9) which for
practical calculation are expanded into sums of small intervals

and i) = ^^' JA + ^'^ft + ... (IX.13)

For this it can be assumed as a first approximation that the dynamic depths and the
pressures are expressed by the same values as valid for the geometric depths. This
first approximation already gives sufficient accuracy in most cases. The most detailed
tables for the calculation of these values are those given by Bjerknes and co-workers
(1910). In these tables it is assumed

Ps, t, D = Pzb, 0, i> + ^s + ft + ^s, t + ^s. D + ((,D

and in addition to the basic values for the homogeneous oceans (Table 1 1 0) six tables are
also given for the numerical determination of the six terms on the right-hand side.
One term, e^, ;, />, is usually so small that it does not have to be taken into consideration.
Hesselberg and Sverdrup (1914-15) have simplified the calculation of the density
in situ by introducing the value of ot, which is known from

Ps,uo=l -}- 10-=^ a,.
Putting

P35, 0,D= P35. 0. 0 + ^ D

gives

Ps./.o= 1 + 10-^c7,.fl, (IX. 14)

where

(^t,D 10-^ = (Tt \0-^ -f ej) -i- €,,D + et,D.

If Gt is known then only three tables are required instead of six for the calculation of
the density in situ.

The calculation of the specific volume and especially the specific volume anomaly
can be simplified in the same way (Sverdrup, 19336):

««, /. p = a35. 0. p + S. where S = 8, + 8^ + 8,, < + 8,, p + 8^, ^

Putting

S. + 8, + 8„, =A,,t
gives

8 = J„, + S„p+ 8,.^. (IX.15)

The first term can be readily found from ct^ and then

a. X 10-3

A..,= \

I + Gt X 10-

310 The Geophysical Structure of the Sea

The numerical value of 035,0,0, is 0-97264 so that

J„ , = 0-02736 - -, "!' ^ ^^in . .
*' * 1 + (^< X 10-3

The specific volume anomaly S can then be determined quickly and easily using three
small tables.
The pressure /? at a dynamic depth D is by definition

CD
/? = Ps,t,D

dD.

Table 111. Example of the dynamic evaluation of oceanographic observations of a single

station
("Meteor" St. 267, 18.11.27; (p = 13-7° N., A = 19-8° W., 4206 m)

Depth Pressure

Temp.

Salinity

Density

10^/1 ,,«

lO^S,,,

10^8*..

lO^S

(m)

(dbar)

(°C)

(%o)

{od

(dyn.m)

21-20

35-236

24-64

331-3

331

0-0827

21-12

35-2I5

24-645

330-8

—

1-0

332

0-0827

00651

16-23

35-57

2615

187-6

1-6

189

0-0434

0-1478

16-25

35-425

26-48

156-3

158

01912

00350

100

13-52

35-355

26-58

146-8

0-1

2-7

150

0-0725

0-2262

150

12-58

35-255

26-69

136-4

0-1

3-8

140

00650

0-2987

200

11-78

35-16

l&ll

128-8

0-1

4-8

134

0-1270

0-3637

300

10-62

3506

26-93

113-6

0-0

6-8

120

0-1160

0-4907

400

10-22

35-14

27-04

103-2

0-1

8-7

112

0-1080

0-6067

500

906

35 02

27-14

93-7

0-0

9-9

104

0-1000

0-7147

600

8-32

34-98

27-23

85-2

-0-1

11-2

0-0920

0-8147

700

7-23

34-89

27-32

76-6

-01

11-5

00850

0-9067

800

6-68

34-88

27-39

700

-0-1

12-2

0-9917

0-0795

900

5-92

34-83

27-45

64-3

-0-2

12-7

0-0740

10712

1000

5-51

34-85

27-52

57-7

-0-2

13-0

0-1320

1-1452

1200

4.99

34-92

27-63

47-3

-0-1

14-0

0-1160
0-1050
00960
0-0900

1-2772

1400

4-68

34-975

27-71

39-7

-0-1

15-5

1-3932

1600

4-14

34-975

11-11

340

-0-1

15-8

1-4982

1800

3-74

34-97

27-81

30-2

-01

16-2

1-5942

2000

3-44

34-965

27-84

27-4

-01

16-6

1-6842

0-1100

2250

3-21

34-955

27-85

26-4

-01

17-4

01100

1-7942

2500

3 02

34-945

27-86

25-5

-0-2

18-2

0-2175

1-9042

3000

2-73

34-93

27-875

24-1

-0-4

19-4

0-2175

2-1217

3500

2-51

34-90

27-87

24-6

-0-5

20-4

0-2250

2-3392

4000

2-37

34-89

27-87

24-6

-0-6

21-6

2-5642

The Geophysical Structure of the Sea 3 1 1

Replacing Ps, t, o by the relation (IX. 14) gives

p = D -\- 10-3 f ^^^ dD (IX.16)

Here only the last term requires numerical integration and has to be summed only to
the depth at which the pressure is required. The anomaly in dynamic depth AD for
given pressures is also obtained in the same way. Since

D = Dss. 0. V + ^ A

8 dp. (IX. 17)

If S is known it can also be found by numerical integration. Using the tables given by
SvERDRUP (1933Z)) the complete dynamic calculation of the values for an oceanographic
station down to 5000 m can with a little practice be done in less than half an hour since
the numbers in the tables are always small.

The absolute values for the specific volume and of the dynamic depth can be ob-
tained by adding the anomalies to the standard values for the standard ocean at 0°C
and 35%o ; they are given in Table 1 10. If o-^ is known accurately to the second decimal
place, then the table will give the density in situ crs,t,D correct to the second decimal
place, and the pressure for a given dynamic depth correct to the third decimal place.
The specific volume anomaly and that of the dynamic depth at given pressures can
be found accurately to the fifth and fourth decimal places, respectively, but the last
two places in the anomaly of the dynamic depth have only computational
significance.

Table 111 shows as an example the complete dynamic evaluation for the "Meteor"
station 267 (18.11.1927; cf^ = 13-7° N., A = 19-8° W., 4206 m), and also the calculation
of the specific volume anomaly and that of the dynamic depth at given pressures in
decibars according to the simpUfied method of Sverdrup.

Chapter X

Forces and their Relationship to the
Structure of the Ocean

1. External, Internal and Secondary Forces

(a) Of the external forces that give rise to or maintain the ocean currents, the most
important are the air currents, the changes in atmospheric pressure at the surface of
the sea, and the periodic tide-generating astronomic forces. These forces can also
initiate water movements in a homogeneous sea. The changes in atmospheric pressure
are transmitted through the entire mass of water down to the sea bottom and thus give
rise to horizontal pressure differences and the formation of gradient currents. The
effect of air currents is twofold: First, the tangential force of the wind on the surface
of the sea (wind stress) produces a surface current which is transmitted by the effect
of viscosity (turbulence) to the water layers beneath the surface. Secondly, the wind
produces waves at the surface of the sea and the pressure exerted by the wind on the
windward side of these waves also initiates water movements in the direction of the
wind (wind drift). These currents produced by the wind and by the changes in at-
mospheric pressure are considerably modified by the deflecting force of Earth rotation
and by the boundary surfaces of the sea (coasts, continental slopes and sea bottom).
The piling up of the water by coasts (Anstau) is by far the most important effect of
the external forces and is responsible for the formation and the maintenance of an
oceanic circulation in the deeper layers of the ocean.

In a sea of homogeneous structure the external forces can produce no change in the
physical stratification of the water mass. In a non-homogeneous sea, however, the
water movements displace different types of water relative to each other, and thus
either directly or due to the boundary conditions produce changes in the thermo-
haline structure of the ocean. This upsets the system of internal pressures forces and
give rise to ocean currents.

(b) The internal forces arise from the vertical and horizontal distribution of mass
within the ocean. These differences in the mass distribution (in horizontal and vertical
direction) are the consequence of changes in the heat content (temperature) and in the
salinity. If at first the water masses are in an internal equilibrium state, this equilibrium
can be disturbed by changes of this type, thereby initiating ocean currents which in
turn tend to restore the system to a new equilibrium. The principal sources for dis-
turbances in the mass distribution can be found at the surface of the sea, where solar
and atmospheric radiation and outgoing radiation first influence the ocean, and where
evaporation also takes place. At the other boundary surface of the sea (the sea bottom)
the intensity of the disturbances is small and usually of no importance in changing

312

Forces and their Relationship to the Structure of the Ocean 3 1 3

the distribution of mass. Within the sea, the turbulence in the moving water masses
may presumably also produce changes in the physical-chemical structure of multi-
stratified water bodies. All these disturbances are, however, small compared with the
changes in mass distribution due to atmospheric influences effective at the sea surface.
The only internal force dependent on the mass distribution is the gradient force.
This force per unit volume is given by the pressure gradient G (see equation (IX. 10)).
The pressure force per unit mass can be obtained by multiplication with the specific
volume so that

r ^P ^ "^P (X n

dn p dn

The pressure field also determines the field of force per unit mass, since the normal to
the isobaric surface gives the direction, and the thickness of the isobaric unit layers
gives the intensity of the pressure gradient at any point in oceanic space.

Bjerknes (1900) by analogy with the pressure gradient introduced a "mobility
vector" B which gives the variations in specific volume in the direction n of increasing
specific volume perpendicular to the isosteric surface.

5 = ^. (X. 2)

The degree of concentration of the dynamic isobaths on an isobaric surface is of
course also a measure of the gradient force, and is at the same time also a measure
of the potential energy stored in the mass distribution. Figure 133 presents a section
through an ocean and two oceanographic stations are indicated by A and B (L km
apart). As a first approximation the pressure surface can be regarded as horizontal
and coincident with the surfaces of equal geometric depth. The surfaces of equal
geopotential (equal dynamic depth) are inclined relative to these so that the same
pressure Pn at dynamic depth Da at station A is found at the greater dynamic depth D^
at station B. Di, — Da is then the difference in potential energy between A' and B'.
This potential difference can be regarded as a force along L which, if present alone,
would set the water masses in motion. The force per unit mass resulting from the
internal pressure difference is then

K = ^' ~ ^°. (X. 3)

According to (IX. 12), D^ — Da is the number of solenoids enclosed within the cross-
section between the two stations A and B from sea-level to the depth in question. This
number per unit length is thus a measure of the internal force resulting from the mass
distribution.

(c) Among the secondary forces are included all those apparent forces that in them-
selves do not give rise to a current but which, when motion is present, are of decisive
importance in determining the final form of the water displacement. These include
the deflecting force arising from the rotation of the Earth (the Coriolis force) which
affects solely the direction of the water movement, the viscosity (boundary fric-
tion and turbulence) which affects more the velocity of a current, and finally the
centrifugal force, which for motion along a curved path (velocity V, radius of

314

Forces and their Relationship to the Structure of the Ocean

ture R) gives a force V^IR away from the centre of curvature; since for an angular
velocity Q,V =^ QR the centrifugal force for unit mass will be Q^R.

Fig. 133. Cross-section through an ocean. A and B are two oceanographic stations.
Full lines: isobaric surfaces (p = const.); Dashed lines: surfaces of equal dynamic depth
(D = const.); The pressure surface p„ appears in station A at the dynamic depth Da, in
station B at the dynamic depth D^ ; L denotes the horizontal distance of both stations

(schematically).

(a) All observations on the rotating Earth are usually made with reference to a co-
ordinate system rigidly connected with the Earth and therefore rotating with the
Earth, although in the classical mechanical sense this is not a permissible reference
system. Such a system should not follow the rotation of the Earth, but would for
example have to be assumed at rest relative to the fixstars (absolute system). If the
basic principles of Galileo-Newton mechanics are used and applied to the rotating
Earth, deviations will appear which are due solely to the movement of the reference
system imposed by the rotating Earth — a fact which we simply have to accept. These
deviations have the character of two apparent forces which are additional to those
forces present in the absolute system.

One of these forces depends only on the geographical location ; this is the ordinary
centrifugal force due to the rotation of the Earth ojV (oj is the angular velocity of
rotation of the Earth — one total revolution per one sidereal day = (27r)/(86,164 sec) =
7-29 X 10~^ sec~^, r is the distance from the axis of rotation of the particle under
consideration). Since this additional force acts both on a stationary or on a moving
mass particle it can be combined with the gravitational force and becomes in this way
part of the force of gravity.

The second force, however, depends both on the geographical location and on the
velocity of the mass particle set in motion on the Earth. This is denoted to CorioUs
force and as a vector acting on unit mass has the form

g = 2[b tu].

(X.4)

Its absolute value is

C = le| == 2Kcosin(t) to)

and it is directed at right angles to the direction of the velocity vector b and to the
angular vector of the Earth's rotation to, which is in the direction of the Earth's axis,
so that I to| = to. It therefore acts at right angles to the tangent to the path of movement

Forces and their Relationship to the Structure of the Ocean 3 1 5

as well as in the direction of the equatorial plane. A complete derivation for the
Coriolis force, unobjectionable in every respect, has been given by Bjerknes and
co-workers (1933). If at any point on the surface of the Earth a co-ordinate system is
chosen with the (A:_y)-plane coinciding with the tangential plane to the Earth (x-
positive to the east, j^-positive to the north, z-positive towards the Earth's interior),
the components of the Coriolis force acting on a material particle on the Earth moving
in any direction with a total velocity V (ii, v, w) can readily be calculated using equa-
tion (X. 4). This gives the components in the three co-ordinate-directions

Cx — 2ctjr sin </> — 2ww cos (f), Cy = —2wu sin 0, C^ — —Icou cos ^. (X. 5)

From these it can be shown that every movement in the tangential plane to the surface
of the Earth will be deflected by the Coriolis force to the right in the Northern Hemis-
phere and to the left in the Southern Hemisphere. The terms cum sole and contra
solem suggested by Ekman can be used respectively for rotation in the direction of the
azimuthal movement of the sun, i.e., to the right in the Northern Hemisphere and
to the left in the Southern Hemisphere {cum sole), and for rotations in the opposite
direction to the azimuthal movement of the sun, i.e. to the left in the Northern
Hemisphere and to the right in the Southern Hemisphere {contra solem).'\ Thus every
movement in a horizontal direction is deflected cum sole by the Coriolis force. It also
follows from equation (X. 5) that there is a vertical component of the deflecting force
only for zonal movements (.v-component u) and not for meridional movements. The
importance of the vertical component for the dynamics of moving masses is quite
small since it acts in the same direction as gravity relative to which it is vanishingly
small. ^

The horizontal component is very important, however; at the poles (<^ = 90°) it
amounts to 1-46 x 10~^ cm sec"^ for m = 1 cm sec"^ and is thus of the same magnitude
as other forces acting in the same direction (gradient forces, tidal forces); it is zero
at the equator and reaches the above maximum at the poles. Since it acts at right angles
tothedirectionofmovementitisunabletoproducechangesin velocity and is incapable of
doing work; it can only produce changes in the ^//-ecZ/o/iofmovement, but these changes
are of decisive importance for the finally established patterns of motion. Due to the
effect of the Coriolis force a mass particle moving freely in a horizontal plane with a
velocity V will follow a curved track. Since the deflecting force acts at right angles to
the velocity (apart from the effect of change in latitude) and its absolute value is
constant, this path will describe a circle which is known as the circle of inertia. The

t Another terminology uniform for both hemispheres is that customary in meteorology: cyclonic
= contra solem and anticyclonic = cum sole.

X The usual statement, that the vertical component of the Coriolis force need not be taken into
consideration, since it is small by comparison with the gravitational acceleration is not entirely correct.
In the static equilibrium state of the sea, gravity and the vertical pressure gradient neutralize each other.
However, under quasi-static conditions the difference between gravity and vertical pressure gradient
is so small that it may be of the same order of magnitude as the vertical component of the Coriolis
force. Nonetheless, it is customary to neglect the latter in calculations ; it can be regarded as an increase
or decrease of gravity so that the acceleration towards the centre of the Earth is now g + Imi cos (p.
It can also be regarded as causing a small change in density in the ratio {g + Iwu cos (p) : g.

At the equator, when m = 30 cm sec-^ it may amount to 5 units in the sixth decimal place in p
or 5 units in the third decimal place in at, which can be disregarded.

316

Forces and their Relationship to the Structure of the Ocean

radius of this circle follows from the condition that the forces acting on the moving
mass particle must balance each other (centrifugal force = Coriolis force).

so that

J/2

2a>Ksin ^,
V

(X.6)

2ca sin (j)

The term "circle of inertia" has been chosen to indicate that relative to the movement
of the rotating Earth this circular movement in a certain sense replaces the linear
inertia of absolute motion. The radius of the circle of inertia is a function of the latitude
and the velocity. Table 112 gives values for this functional relationship.

Table 1 1 2. Radius of inertia circle as a function of V and </>
(V in mm sec-^, cm sec-^, m sec-^: R in m, 10m, km units)

0 . .

5°

10°

20°

30°

40°

50°

60°

70°

80°

Poles

V= 1

157

236

118

315

158

393

198

100

472

237

120

551

277

140

629

316

160

110

708

356

180

124

786

395

201

137

106

The time required for one complete rotation on the circle of inertia {inertia period)
is given by

IttR it 1 2 sidereal hours

T =

CO sin (f)

sin <f>

If the period of rotation of the plane of oscillation of a Foucault pendulum is a pen-
dulum day = {2tt)1{o) sin </•), the period of rotation of the circle of inertia will be a
half pendulum day whatever the value of the velocity V. Table 1 1 2a shows the time re-
quired for one revolution on the inertia circle (the inertia period) at different latitudes
(from 5 to 5 degrees).

Table 112a. The period of rotation of the inertia circle {inertia period).

4. . . . .

0°

5°

10°

15°

20°

30°

40°

50°

60°

70°

80°

Poles

Hours (^ pendulum day)

138 69

18-7

15-7

13-9 12-8

12-2

120

In lower latitudes the period may be several days, in middle latitudes 24 h and at
high latitudes half a day; here, since it has a similar period as the daily and half-daily

Forces and their Relationship to the Structure of the Ocean 317

components of the tide-generating forces, it is of particular importance in the dynamics
of periodic phenomena.!

{P) In addition, /r/c?/o« is also of considerable importance in all oceanic movements.
Like all liquids, sea-water has a viscosity which for deformation manifests itself as an
internal friction. The friction of water over a solid and rough sea bottom is primarily
an external boundary surface friction. This type of friction represents a retardation of
the flow of the current but only in relatively shallow water can be taken as a measure
of it; the most simple assumption describing the frictional mechanism is that the
gliding flow of the water over the solid bottom meets a tangential resistance which is
assumed proportional to the velocity of the current V. The frictional force in this case
would correspond to a vector with a direction opposite to that of the velocity vector
and has the absolute magnitude kpV, The quantity k is termed the coefficient of
gliding friction. Hydraulic investigations on the dissipation of the kinetic energy of a
river due to friction on the river bed have shown that the frictional force per cm^ of
the bottom surface is proportional not to the first power but rather to the square of the
flow velocity. It can be expected that the dependence of the boundary surface friction
on the velocity will also be of the same kind for shallow ocean currents. Taylor
(1920) attempted to apply the conditions found in natural channels to coastal oceanic
currents in shelf areas. In the friction formula the coefficient k for a normal sea bottom
has the value 0-0026 for depths of about 50-100 m so that

R = -2-6 X \{)-^pV\ (X.9)

At more shallow depths with an especially irregular sea-bottom topography k may
increase considerably (100 times the above value or even more). These frictional as-
sumptions refer always to the mean tangential resistance exerted over the whole of a
column of unit cross-section from the bottom to the sea surface due to the eff'ect of
boundary friction at the bottom. However, these assumptions do not specify the nature
of the friction in the interior of the total water column above the sea bottom. The
internal friction appears as a tangential shearing stress r between individual layers of
water gliding one above the other with different velocities. This stress per unit area is
proportional to the velocity gradient perpendicular to the direction of the flow
dVjdn, so that

T=)Lt^-. X.IO

The quantity ju is the coefficient o^ dynamic viscosity and has the dimensions [g cm^^
sec"^].J

t The inertia movement has the form of a circle only if the Coriolis force is constant (mostly
assumed as the mean value for the meridional width of the inertia circle). A general derivation for
varying latitude has been given by Wipple (1917) but this was confined, however, to movements near
the equator, since sin (f) was replaced by the arc 4> of latitude and cos (^ by 1. Inertia movements super-
imposed on horizontal and zonal currents play a large part in the dynamics of ocean currents especially
the occurrence of long waves and in vortical disturbances. (See in this connection Defant (1956) and
Vol. I, Pt. II, Chap. XIII, 6.)

X The origin of viscosity can be sought in the continuous equalization of velocity between super-
imposed layers of water gliding over each other in a moving water mass. This equalization is due to
the interchange of individual molecules and the consequent transfer of velocity from one layer to the
next. This viewpoint is, however not entirely correct since the molecules in a hquid are so closely packed
that usually they can only oscillate within the small intermolecular spaces present and therefore only

318

Forces and their Relationship to the Structure of the Ocean

For this assumption concerning the inner friction, the effect of the solid, stationary
sea bed appears as a corresponding boundary condition. If n is the direction of the
normal to the sea bottom (z = 0) then

(1) for completely frictionless movement of the water over the sea
bed (r = 0): dVldn = 0;

(2) if the water is stationary at the bottom (z = 0): K = 0;

(3) for part-time gliding at the sea bottom, that is for a discontinuity
of the velocity at z = 0: dVjdt = f(V), where /(K) is a certain
function of V, for example, kpV^.

In a volume element 8x 8y 8z (see Fig. 134) in a current in which the velocity V
in a direction perpendicular to the vertical direction z is very much stronger, there
will be a shearing stress rSxSy on the lower surface 8x8y and a corresponding

- (X. 11)

Fig. 134. Computation of the frictional force from the shearing stresses.

^T j^ {8Tl8z)8z}8x8y on the upper surface at a distance Sz from the lower. On the
entire volume element there acts thus a frictional force (8Tldz)8x8y8z so that accord-
ing to (X. 10) the frictional force per unit mass in the direction of the x-co-ordinate

will be given by

fi 8^V

8z^ '

R:r. —

(XJ2)

Where fx can be regarded as a constant.

From the general theory of friction in hquids it follows that for an incompressible
fluid (and thus also with sufficient accuracy for sea-water) the components of the fric-
tional force per unit mass in a viscous liquid are given by the three expressions :

u ix M ,

;?^ = - An, Ry = ^ Av, i?, = - Aw,
PR P

(X.13)

Footnote continued from p. 317

seldom change position. These occasional changes in position are facilitated by the action of a tan-
gential shearing stress especially in the direction of the stress itself and this alone permits the individual
layers to glide over each other. The more frequent the changes in position of the molecules, the lower
is the internal friction (viscosity) characteristic of the liquid.

Forces and their Relationship to the Structure of the Ocean

319

where ii, v, w are the velocity components in the direction of the three co-ordinate
axes and A is the Laplace operator 8^l8x^ + 8^l8y^ + 8^l8z^. The quantity v = ixjp
is called the kinematic viscosity coefficient and has the dimensions [cm^ sec"^]. For
numerical values of ju and v for pure water and for sea-water see Vol. I, Pt, I, p. 104.
The actual movement of water masses in the oceans does not correspond to a
simple ordered gliding of the individual superimposed layers relative to each other,
but is rather a random disorganized movement that takes place in vortices and rolls
similar to those which can be seen in a smoke plume. The first type of motion is called
layered or laminar and the second turbulent. In turbulent flow there is a transfer of
the flow momentum from one layer to another, not by the interchange of molecules
as in physical internal friction but by the exchange of large elements of water (eddies)
which move rather irregularly back and forth between the diff'erent layers and thus
bring about a reduction in the velocity in the direction of the basic current; this is
then referred to as virtual internal viscosity or eddy viscosity, which in an analogous
way to the molecular viscosity can be characterized by a special eddy viscosity coeffi-
cient. It is easily seen that the eddy viscosity, by its nature, will be more eff'ective than
the molecular viscosity and is also understood by the numerically much larger viscosity
coefficients. However, apart from this, the turbulent coefficient is no longer an in-
variable quantity like the molecular viscosity at constant temperature, but depends on
the nature and the intensity of the turbulence itself. Further, the components of the
frictional force of turbulent viscosity can be expressed in exactly the same way as those
in equations (X. 12 and 13) if ju, is replaced by the turbulent viscosity coefficient rj.
If this is not constant then, for example, equation (X. 12) is replaced by the expression

1 a

p 8z

('£)

X.14

and the same applies for the other expressions in (X. 13).

To a very large extent ocean currents are movements along quasi-horizontal planes
so that the turbulent viscosity for small oceanic spaces is limited to that appearing in
connection with layered gUding motion of the water masses. Within the moving water
mass turbulence creates a definite vertical velocity profile and tends to maintain it.
If there is no viscosity this profile must be linear (see Fig. 135, 1). The velocity of the

Fig. 135. Main types of vertical velocity distributions: (I) in the case of no friction; (II) in

the case when friction retards the mean current filament; (III) in the case when friction

accelerates the mean current filament; (IV) for a constant frictional force.

filament (a) in the middle of the current is then the mean of the velocities of the ad-
jacent upper and lower masses. The accelerating influence of the upper layer will be
exactly compensated by the retardation at one of the lower. In case II, where the

320 Forces and their Relationship to the Structure of the Ocean

velocity of this middle layer is greater, the adjacent layers will exert, due to the
transfer of their flow momenta, a retardation on the current maximum in the middle
and will eventually eliminate it. The middle layer in case III will be accelerated by the
equalization of velocity in the turbulent flow. Equation (X. 12) also shows that for a
constant internal friction the vertical profile must take the form of a parabola.

2. The Basic Hydrodynamic Equations

For a complete description of the water movement in ocean currents, it is necessary
to know on the one hand the path of each small element of water in it, and on the other
hand the position of such a small element along this path at any time; i.e. it is neces-
sary to know the co-ordinates of a small element of water as a function of time. The
basic hydrodynamic equations of motion in their most general form are the mathe-
matical-physical tool for dealing with and for a theoretical understanding of the
different successive states of a water mass.

The motion can be looked at from two different view-points. The different mass
elements may be followed as they pass a. fixed point in space and particular attention
may be paid to the changes in the state of motion of the water mass which occur at
this point. Alternatively, the changes of state of individual small elements moving
along their track may be followed, and thereby a description of the conditions in the
current in the course of their displacement can be obtained. The first approach gives
the Eulerian basic hydrodynamic equations of motion (Euler, 1755) and the second
leads to the equations of motion of Lagrange (Lagrange, 1781); both of these con-
cepts are applied in oceanography according to the type of problem to be solved.

For a small element of water in the point .v, y, z the components of the velocity are
denoted u, v, w in the directions of the co-ordinate-axes of a left-hand system (xy-
plane horizontal, x-axis positive to the east, >'-axis positive to the north, z-axis posi-
tive towards the centre of the Earth), They will be functions of x, y, z and of the time
/. First of all the basic Newtonian relationship of mechanics is applied:

Mass X acceleration = sum of all forces.

The individual accelerations dujdt, dvjdt and dwjdt are made up of two parts. The first
part arises from changes in the state of motion at the point; it is given by dujdt,
dvjdt and dwjdt (local change). The second arises since after a small time dt the water
elements under consideration are no longer found at the initial point (.v, y, z) but are
displaced by udt, vdt and wdt, respectively (advective change). Thus to the local part
must be added an advective part, so that the total individual acceleration in the x-
direction of the small elements of the liquid under consideration will be

du du du du du

Similar equations apply for dvjdt and dwjdt. It may be emphasized here that the partial
derivative djdt always represents the change in the quantity under consideration at a
fixed point, while the total derivative djdt represents the individual change in a quantity
for one and the same element (which changes its position with time).
Taking the mass of unit volume as p, and considering that since the only external

Forces and their Relationship to the Structure of the Ocean

321

conservative force is gravity acting in the positive direction of the z-axis (downward),
the pressure gradient forces will be given by

\ dp I dp I dp

p dx* P dy* p dz '

respectively, and introducing the CorioHs force according to (X. 5) and the frictional
forces according to (X. 13), then the basic hydrodynamic equations of motion will
take the complete form

(X.16)

du
dt~

du du du du . I Sp fi ^

w: + u ^ + V ^+w -^ = + 2wv sin 0 ^ + -Au,

dt dx dy 8z ^ p dx p

dv
dt~

dv dv dv dw ■ . ^ ^P H- .

8t dx oy dz p dy p

dl^

dw dw dw dw \ dp a

-77+ u -^ + V -^ + w w- = g — 2wu cos (^ — + - Z

dt dx dy dz ^ p dz p

The third equation in the 2-direction can be considerably simpUfied, which shall be
done at once. Since the movements of the water in the ocean occur very largely in a
horizontal plane and w, dw/dt and the frictional term in this direction can always be
assumed to be small, and further, since the vertical component of the Coriolis force
can be neglected, the third equation in (X. 1 6) reduces to

1 dp

(X.17)

0=^

which corresponds to the basic hydrostatic equation (see p. 337).

For problems involving the whole or an extended part of the rotating Earth it is
convenient to use polar co-ordinates. The reference surface selected is the free sea sur-
face in a state of equilibrium (usually it is sufficiently accurate to take a spherical
surface with the mean radius R of the Earth) and as co-ordinates can be taken the pole
distance {^ = 90 — (j), the longitude A and the distance z from this surface (along the
radius of sphere R, positive outwards). The velocities relative to the Earth along the
three axes are then

u = (R + z)

d&
dt

V = Rs'm >&

dX
It

and

dz
Jt

(X. 18)

If the external forces have a potential Q-f and if the frictional terms are omitted, the
equations of motion take the following form

du ^ - ^ d /p \

2ojv cos §■ — —

R + z dd'\p

dv
dt

dw
df

+ 2<ou cos 'd' + 2cjL}W sin d' =

R sin

1 8X

— 2cov sin '&■ =

+ Q

(X.I9)

t The forces X, Y, Z have a potential Q when they can be represented by

ei? _dQ _8Q

~dx ~ ~dy ~ ~dz

X =

322 Forces and their Relationship to the Structure of the Ocean

Since the depth of the sea is always very small as compared with the dimensions of the
Earth, the term {R + z) in the first equation of (X. 1 8) can be replaced in good approxi-
mation by R.

In the Lagrange equations of motion the co-ordinates x, ^, z of a small mass element of the liquid
are viewed as functions of the independent variables a, b, c and of the time t; a, b, c are the initial
co-ordinates of the particle under consideration, i.e., values of x, y, z at the time / = 0. These functions

;c = /i (a, b, c, t), y = fz (a, b, c, t), z = f^ (a, b, c, t),

thus describe the history of each small element of the liquid vi'ithin the current. If only the time t
is altered, they give the path of the element under consideration; if on the other hand t is constant
and only a, b, c are allowed to change, this gives the positions of the different elements at one and the
same instant of time. Since the accelerations of the element a, b, c at the time / are given by

du _ d^x dv d^y dw _ d^z
'dt ~ dt^' ~dt ^ dl^' IJi ~ dF^

the equation (X.15) can also be written in another form

^ - X= --^ ^^' _ y = _ ' ^ ^!' _ Z = - 1 ^^
dt^ P dx dt^ p dy dt^ p dz'

To eliminate at the right-hand sides the derivatives with respyect to .v, v, z these equations can be
multiplied at first with

dx dy dz
8a 8a 8a

then with

dx dy dz , dx dy dz

-7> T' —,■ 3nd --, ^, ~,
db db db dc dc dc

respectively, and finally can be added. If the forces have a potential Q, the Lagrange form of the equa-
tions of motion is obtained

Idhc \ 8x id^ \ a V IdH ^Y^ ,^ ^P_^

\dl''~ ^) aa + U/2~ ^ ) da + \dl^~^)da^~p 8'a~ ^'

\dt^ J db^\dt^ I db ^ W/2 ^ I 8b^ p db

Id^x \ dx id^y \ dv (d^z \ dz I 8p

b/2 - ^} e-c + \d^ - ^ } Tc^ \dt^~ ^ } 8c^

p cc

The hydrodynamic equations of motion form a very complex set of equations. They
have to be solved in order to obtain a complete description of the state of motion but
only in very rare and in the most simple cases it is possible to arrive at a final and
definite solution. In most cases it is considered sufficient to determine, if possible, the
state of motion at each place and at each time without paying any attention to the
further history of the individual water elements. There is a considerable simplification
possible when dealing with so-called stationary currents. These are currents in which
the state of motion at each point does not change with time and is thus completely
fixed by specifying its direction and velocity. The condition for a steady state in the
current is thus

8u dv dw

8", = a, = a? = 0- <'^-2«>

Forces and their Relationship to the Structure of the Ocean 323

Some kinematic properties of the motion should perhaps be referred to here. The
path of a small water element is obtained from the three simultaneous equations :

dx = udt, dy = vdt, dz = wdt. (X.21)

The integration constants for f = 0 are then the three initial co-ordinates a, b, c of
the water element under consideration.

The instantaneous state of motion in a water mass is given by the stream lines
(see Chap. Xll) which everywhere indicate the direction of a current by the tangent at
the point under consideration. Their differential equations are

dx dy dz

— - — = —. (X. 22)

U V w

Since the state of motion in a steady current does not change with time it is under-
standable that the stream lines in this case coincide with the trajectories of the water
elements. Steady currents are not without accelerations since only the local part of
the acceleration disappears; the advective part, for example, u(duldx) + vidujdy) +
w{8ul8z) requires that the moving water element reaches any point with a velocity
equal to that prescribed for that point,

3. The Continuity Equation and the Boundary-surface Conditions

To the equations of motion must be added, as a special condition, the continuity
equation which is based on the law of the conservation of mass. This states that in any
volume element specified in the interior of a liquid the mass entering it at a given time
must be equal to that leaving it at the same time. Any excess in one or the other direc-
tion must appear as a corresponding change in the density if the liquid will permit such
a change. Taking a volume element SxSySz, investigation of the extent by which, as a
consequence of flow through the boundaries the amount of liquid enclosed in it
varies, shows that for a conservation of mass the continuity condition is given by the
equation

dp dpu dpi) dpw

Using the relationship equation (X. 1 5) this can be given the following form

\ dp ] da 8u 8v 8w

p dt a dt 8x cy 8z

In an incompressible liquid dpldt = 0 the continuity equation reduces to

8u 8v 8h'

^ + ^ + — = 0 - (X.25)

ex oy oz ^ -^

This does not assume that the liquid has the same density everywhere (homogeneous
medium). The expression cujdx + cvjcy + bwj8z indicates the volume increase in
unit time per unit volume of the element and is usually termed the three-dimensional
or total divergence of the vector (m, r, vv). The continuity equation for an incom-
pressible medium is then

div (//, r, h) =-- 0. (X.26>

324

Forces and their Relationship to the Structure of the Ocean

Since the rotation of the Earth does not affect the conservation of the mass, the con-
tinuity equation does not contain the angular velocity of the Earth's rotation when a
polar co-ordinate system is used for the rotating Earth (co-ordinates: pole distance
'& — 90° — (f), longitude A and r along the Earth's radius R). However, there are
changes in the cross-section of a current for meridional motion due to the convergence
of the meridians and for vertical displacements of mass due to the divergence of the
Earth's radii. Thus in the continuity equation for polar co-ordinates, in addition to
the previous terms derived from flow through the volume element, there will be two
further terms considering these further circumstances in this special co-ordinate
system. These give the following equation :

dp 1

87 '^ R sin ^

dpu sin '& 8pv'

dpw 2pw
+ ^ + -^ = 0. (X.27)

The effect of the convergence of the meridians is expressed in the term (puIR) cot g'&
which is obtained by differentiation of the first expression in the brackets and the effect
of the divergence of the Earth's radii is contained in the term 2pwjR. Since for vertical
displacements of mass in the sea, which is shallow relative to the Earth's radius, the
vertical velocities appearing are very small, this last term is not too important and can
safely be neglected. For small oceanic spaces the convergence of the meridians can also
be disregarded in first approximation, though not for large-scale ocean currents
(see Chap. XXI).t

If the liquid has boundary surfaces either at a solid body (the sea bottom) or when
it is surrounded by differently stratified liquids (other water bodies) the continuity
equation will take special forms and must be replaced or supplemented by special
boundary conditions. At a solid boundary, in order to secure a reasonable state of
motion with no empty spaces, the component of the velocity perpendicular to the
surface must be zero. If /, m, n are the direction-cosines of the normal to the surface
then a necessary condition is

lu -{- mv + nv — 0.

(X.28)

t The continuity equation which corresponds to the Lagrange equations of motion is more
difficult to derive and reference should be made to text-books of hydrodynamics. Taking the functional
determinant

8(x, y, z)
8{a, b, c) '

the condition of constancy of mass in a volume element 8a 8b 8c will be

8(x, y, z)
d{a, b, c)

where Po 's the initial density at the point {a, b, c). For incompressible liquids p = Po the continuity
equation takes the form

8{x, y, z)
8(a, b, c)

= 1.

Forces and their Relationship to the Structure of the Ocean 325

At all inner boundary surfaces, on the other hand, the velocity component perpendicu-
lar to the boundary surface must be the same on both sides of the surface. If the values
for the quantities on both sides of the boundary are specified by separate indices,
then this kinematic boundary condition can be represented as a special case of equation
(X. 28)

/("i - «2) + rn{vi - ^'2) + n{yv\ - w^) = 0. (X.29)

From the point of view of continuity it is allowed to make a free choice about the
velocity component parallel to the inner boundary surface and solid surface, respec-
tively.

If the liquid has Sifree upper surface this will be subject to the condition that all the
small fluid elements which belong to it will always remain in the liquid. If/Cv, y, r, /) =
0 is the equation for the free upper surface the foregoing condition requires that

In addition to the kinematic, there is also a dynamic boundary-surface condition
that must be satisfied at inner boundary surfaces as well as at a free surface. This
requires that at the discontinuity surface where the individual quantities are subject
to sudden changes, the pressure must be the same on both sides of the boundary. If
/(x, y, z, /) = 0 is the equation for the discontinuity surface, which may be either
moving or stationary, and if/7i and/72 give the pressures in the medium on both sides
of the surface as functions of .Vi, y^, z^ and x^, J2, z^, respectively, then the dynamic
boundary condition will require that values of x, y, 2 and t, in order to satisfy
f{x, y, z, t) = 0, must also satisfy the equation

PiiXi, >i, Ti, 0 — p^ix^, J2, Z2, /) = 0. (X.31)

4. Potential Flow, the Bernoulli Equation, Impulse and the Impulse Form of the
Hydrodynamic Equations

In very many cases the velocity components u, v, w can be expressed by a function
9, so that

80 S(D do

This function then is called the velocity potential, and movement for which a function of
' this type is valid has been termed o. potential flow. By this kind of definition it is shown
that if such a potential is present:

(1) The stream lines will be everywhere perpendicular to the surfaces 9 = const,
(equi-potential surfaces of velocity). This follows from (X. 22) when combined
with (X. 32).

(2) The following combinationary relationships :

du 8v dv 8w 8w 8u

8y 8x ' 8z 8y ' 8x 8z

will apply ; these state that the current in the presence of a velocity potential is
irrotational (free of vorticity).

326 Forces and their Relationship to the Structure of the Ocean

(3) The continuity equation for an incompressible medium will take the form

S^cp c^cp 3^9

Neglecting the Coriolis force and the frictional forces, the three Eulerian equations
of motion equation (X. 16), on multiplication by dx, dy and dz, respectively, and
taking further into account the identity

du du I d ^ ^

rf^ = «7 + 2 8Tx("^ + '-^ + "'^) ('^■33)

and by subsequent addition, can be compressed into the single equation

where F(t) is an arbitrary function of t alone and Q is the potential of the external
forces. For a steady current

(8u 8v 8w \
di^ 8i ^ 8t ^^)

in which the stream lines coincide with the trajectories of the fluid elements

U^ + V^ + H'^ p

^ +~+^=C, (X.35)

where the quantity C is constant along each stream line but changes on passing from
one stream line to another. The equation (X. 35) is called the Bernoulli theorem
(equation). It shows that for steady motions the pressure at points along a stream line
is greatest where the velocity is smallest and vice versa. Considering that a fluid particle
on transfer from higher to lower pressure is subject to an acceleration (increase in
velocity) the above statement is readily understood. This is another way of expressing
the conservation of energy, since for unit mass the first term is the kinetic energy of
motion, the second is the work done against pressure and the third is the potential
energy; in a steady flow the sum of these energies along a stream line must be constant.
If the water movement is solely influenced by the gravity force, then since Q = gz,
the Bernoulli pressure equation will have the form

^ + - + ?z = const., with m2 m f2 ^ ^^,2 - c\ (X.36)

2 p

For a two-dimensional potential flow it is convenient to introduce a stream function ifj
defined by the relations

«=-^, v=+^^ (X.37)

and therefore from (X. 32)

Sep Si/» ^9 8ijj

8x^ 8)'' 8y^ ~ 8x'

Forces and their Relationship to the Structure of the Ocean 327

In addition, the differential equation J0 = 0 must also be satisfied by i/-. Since the
curves ^ — const, are perpendicular to the curves 9 = const,

\dx 8y '^ 8y dx "'

They represent stream lines (hence, the name stream function).

It can easily be shown that every analytical function of the complex variable
r = .V + iy satisfies the continuity equation Jcp = 0, i.e. represents a solution for
the equations of motion. If this function is given by

F(z) = F(x + iy),

then its real part is the velocity potential (p and the imaginary part is the stream func-
tion ifj or vice versa. This important consequence allows simpler current systems to be
completely worked out kinematically. Use will be made of this later (see Chap. XII, 3).

In a few important cases the use of the impulse theorems for steady currents in a water
mass has considerable advantages. The product of mass and velocity is termed the
impulse or momentum; as a vector, like velocity, it has three components. The impulse
theorem states that for any arbitrarily limited water mass (the outer boundary sur-
faces all together are usually termed "control surface") the change with time of the
impulse in it is equal to the sum of the external forces acting on the mass. The internal
forces in the system balance each other according to the principle of action and
reaction. The change in momentum can be divided into two parts. The first gives the
change with time of the impulse in the volume under consideration enclosed by the
control surface; for a steady current this term vanishes. The second is the momentum
entering or leaving it in unit time through all the boundaries (total control surface).
For a steady current the vector sum of all pressures acting on the control surface must
be equal to the transport of impulse through it.

As an example, the following two cases will be considered here. Fig. 136a shows a
straight current tube formed by stream lines ; we consider the part between 1 and 2.
At the cross-section 1 (surface F^) the current enters with a velocity V^. The water

Fig. 136a

mass transported in unit time is pV-^F^, the impulse transport (momentum flux)
through Fj into the volume under consideration is p Fj^Fi ; similarly, at cross-section 2
(surface F^ an impulse amount pV^Fz leaves the enclosed space; as a "counter action"
it has to be taken with a negative sign. The impulse amount remaining in the space is
thus p(Ki^Fj — V^F^. In a steady current, in order to secure an equilibrium state,

328

Forces and their Relationship to the Structure of the Ocean

it has to be balanced by the vectorial sum of all the surface pressures, that is, by
Fj/?! — F^Pz- This gives for the current tube the equilibrium equation

Ki2 +

a-i

P2.

K22+-IF2

which corresponds to the BemouUi pressure equation.

If the current tube is curved (Fig. 1 36^) the forces at both places 1 and 2 will have
different directions and the resultant R of the two forces (indicated at point A) shows
the effect of the pressure exerted by the curved flow on the adjacent water masses.

(b)

Fig. 1366

By the introduction of the contmuity equation, the equations of motion can be put
in a form which expresses changes in impulse more clearly {impulse form of the equation
of motion). Multiplying the continuity equation (X. 23) by m, v, w and adding these
expressions respectively to the first, second and third of the equations of motion (with-
out Coriolis force and friction terms, X, Y, Z are the external forces), then

dpu dpuu 8puv dpuw

dx'
dp

8pv 8pvu 8pvv 8pvw

'8i '^ ~8x '^ "ajT "^ ~aF~ ^ ^ 8/ >

(X.38)

8pw 8pwu 8pwv 8pww

"aT "^ "ax" "^ ~e^ "^ 8z

8z'

These show that the changes in the momentum within a volume element can be re-
garded either as the result of forces acting on the mass contained within the volume
element, or as the result of the mass flux passing through the boundary surfaces carrying
its own momentum with it.

The impulse-form of the equations of motion (X. 38) can be used with advantage
in considerations concerning the internal structure of turbulent currents (Reynolds,
1 894). At any point of a turbulent flow there will be more or less strong variations in
the flow velocity. These variations will, however, balance each other completely if
on the average the current is steady, and if a sufficiently long period is considered. The
velocity components at a given point can then be represented by

M = W + «', V = V -\- V', W = W + H'', (X.40)

Forces and their Relationship to the Structure of the Ocean 329

where m, d, w are the mean values of these components and u', v', w' are the compo-
nents of the superimposed turbulent motion for which by definition

u' = 0, V = 0, w' = 0. (X.40)

The bar over these symbols indicates mean values considered over a sufficiently long
time. It should further be noted that the mean values of the squares and products of
«', v', w' of course must not necessarily vanish.

If the impulse equations (X. 36) are apphed to such a turbulent flow it is not suffi-
cient to consider the equations for the mean steady flow alone, since also the turbulent
parts of the velocity changes are involved in the relationship between the mean steady
current and the forces acting on the masses. This can be derived directly from the
impulse theorem. Considering, for example, a part of the "control surface" that is at
one time vertical to the x-axis and at another time vertical to the jv-axis, then in the
first case a mass pu will pass through a unit area in unit time; the impulse transport
due to the x-component u of the velocity is then pun and its mean value over a longer
period puu. Now

uu — {it -\- u'Y + «^ + 2wm' + u'^.

In deriving the mean value uu it should be noted that u is already a mean value of u
and w' = 0, so that

puu — pu^ = pu'^.

To the impulse of the steady mean current a turbulence contribution is added in form
of the square of the turbulent variation in velocity, which when inserted in equation
(X. 38) has the effect on the mean motion of an additional pressure.

Similarly, a mass pv will pass through unit area of the control surface perpendicular
to the >^-axis in unit time. The x-component of the impulse transferred through the
surface is thus, in this case, puv and taking an average gives puv per unit time. With

uv — uv ■}- u'v -\- uv' + u'v',

puv = puv + pu'v'.

In addition to the impulse of the steady mean current puv must be added a turbulence
contribution which in general does not vanish; because positive values of m' are mostly
correlated with positive values of v' and vice versa, so that the products are preferably
positive. In the opposite case the products are mostly negative.

If this turbulent contribution of the impulse transport is transferred to the right-
hand side of equations (X. 36) it can be taken as a force acting along the .v-axis, which
in all cases will be perpendicular to the >'-axis. It can therefore also be considered an
apparent shearing stress

r = -pTv' (X.41)

arising from the turbulence of the current and was previously regarded (see pp. 3 1 7-3 1 9)
as an apparent internal frictioH. Equation (X. 41) mediates between this viewpoint
and the equation (X. 10) which defines the turbulent viscosity coefficient r].

5. Circulation and Vorticity

The Bjerknes theorem concerning the formation of vortices and circulation accelera-
tion (1898, 1900, 1901) has been found very useful in the theoretical treatment of

330 Forces and their Relationship to the Structure of the Ocean

problems arising with oceanic currents. This applies to the dynamics of moving
""non-homogeneous'' media in which the effects of friction are considered unimportant.
This method of treating problems of oceanic movements has the particular advantage
that it takes into account the total ejfect of the mass field on the water movements
including all their smaller details. It can only be used in its simpler form by neglecting
friction; in general, however, at a distance from the boundary surfaces the friction
does not change to any large extent the nature of the currents set up by the internal
forces.

{a) Circulation for an Earth at Rest and for a Rotating Earth

In the presence of (/?, a) solenoids, motions are always initiated the nature of which
is that of a circulation, i.e., motions following in the most simple case a closed path.
In a moving fluid a continuous chain of material elements may lie in a closed curve s.
The velocity component of one of these small elements tangential to the curve s
shall be F<. The sum of all these components along the curve s is defined as the
circulation C of the curve s

C = & Vt ds, (X.42)

where ds is a linear element of the curve s. An expression for the change of C in time is
easily obtained from the equations of motion (X. 1 6) (stationary Earth, frictionless
motion).

(X.43)

Since normally the external forces (gravity) have a potential, the first integral vanishes
and the equation becomes

— = -^ ^ y.dp = N, (X. 44)

where A'^ is the number of isobaric-isosteric unit solenoids, enclosed by the curve s
(see p. 307 equation (IX. 1 1)). Assuming that the curve s lies in a plane, then:

(1) The circulation is constant with time (dCldt = 0) if a is constant over the whole
of the space under consideration (homogeneous sea) or if it is a function of
pressure. The isobaric and the isosteric surfaces then coincide and the mass
distribution is barotropic.

(2) A circulation acceleration will be present if the specific volume is dependent not
only on the pressure but also on other properties of the water (temperature,
salinity). The mass field is then baroclinic. Form equation (IX. 12) for a curve
5 in a dynamic section formed by two vertical lines, the physical sea-level (/? = 0)
and an isobaric line at depth p^, the number of solenoids enclosed will be
given by the diff'erence in dynamic depths of the isobar p^ at the two stations.
This gives

-^ = N= Da- D,. (X.45)

Forces and their Relationship to the Structure of the Ocean
In the two-dimensional case {x, z)

and from Stokes's law it follows that
a dp =

dp
dx + a ^r- dz

r , { C /8a 8p da 8p\

f^'^ = ]\ [8xTz-Tz8-xj

dxdz

331

(X.46)

(X.47)

If now e and /3 are the angles of the ascendent of the pressure 8plcn and the ascendent
of the specific volume 8al8n, respectively, with the .r-axis, then

da da

^ = ^ cos ^, ^ = ^ sin ^,

8x 8n

8p 8p

^ = ^ cos €,

8x dn

^dp .
^T Sin e
en

and from equation (X. 47)

and

adp =^

da dp
dn on

sin (e — P) dx dz

~di

da dp

-^ ■?- sm (e

on dn

iS) dx dz.

(X.48)

(X.49)

The two possible cases are shown graphically in Fig. 136c. If e > j8 then the circulation
acceleration dCldt < 0 and produces an anticyclonic circulation. If, on the other hand,

'1 a'2 a'J

p p*l p+2 p*3 p*4 p*5 p+6

Fig. 136c. Cyclonic and anticyclonic circulation movements for different pressure gradients

and specific volumes.

e < /8 then dCjdt > 0 and the resultant movement is cyclonic. In the two cases (see
Fig. 136c) the circulation proceeds from the ascendent of pressure to the ascendent
of specific volume. In oceanography, as a first approximation, the isobaric surfaces
are horizontal, i.e. e = 90°, and thus

dC
'dt

g da
a dn

cos ^ dz

(X.50)

A cyclonic circulation is present when ;8 > 90° and thus the isosteres decline towards

332 Forces and their Relationship to the Structure of the Ocean

the left relative to the isobars and an anticyclonic circulation will be present when
i8 < 90° and so the isosteres decline to the right.

The circulation theorem gives the change in absolute circulation C, i.e. the circula-
tion referred to a co-ordinate system at rest. For oceanographic problems, however,
it is the change in the circulation relative to the Earth which is of interest. The abso-
lute velocity Va referred to a fictitious Earth at rest can always be represented as the
sum of the relative velocity Vr relative to the rotating Earth and the velocity V^ of
Earth rotation. Thus in the direction of the tangent / to the curve s

Va,t = Vr,t + n.,

and thus

C, = Cr-\- Ce. (X.51)

The circulation Ce can be calculated. If the curve s lies in the equatorial plane then the
velocity Vg for each point on the curve will be cor where r is its distance from the Earth
centre. The component of it coinciding with the direction of the tangent / to the curve
s will be given by

Ve, t= rw cos P,

where )S is the angle between the tangents to the circle r and to the curve s. Thus

Ce, t = 60 r cos ^ ds =^ 2co \ r cos P ds = 2io F, (X.52)

where Fis the area enclosed by the curve s. If the curve s does not lie in the equatorial
plane it can be resolved into its projections on the equatorial plane and on the meri-
dional plane. Since the velocity V^ is perpendicular to the meridional plane it will have
no component in the direction of the tangent to the projection of the curve on the
meridional plane and its contribution to Cg.t will therefore be zero. The contribution
of the projection of the curve on the equatorial plane is identical with equation
(X. 52); F is now the area within the projection of the curve s on the equatorial
plane. Thus for the relative circulation acceleration is obtained

dCr ^ dF .,, ^^^

-^ = N - 2aj -J-. (X.53)

dt dt ^ '

As a first approximation, if the area is not too large, the latitude ^ is assumed constant
and Fcan be put equal to F^ sin <j>,'\ where F^ is the area within the projection of the
curve s on the sea surface. Thus

^ = TV - 2co sin 0 ^. (X.54)

The acceleration is made up of two terms; the first is the number A^ of solenoids en-
closed by the curve and acts always in the direction from the ascendent dajdn to the

t More exactly

dF dF„ . , ^ dtp dFm . , ^ V

-^- = -^- sm (f + Fm cos f ^1= ^i sin (p + Fm cos (p ^.

Here v is the south-north velocity; in middle and higher latitudes the second term is insignificant but
towards the equator the first vanishes and the second becomes important.

Forces and their Relationship to the Structure of the Ocean 333

pressure gradient dpjdn (Fig. 136c); the second represents the product of the CorioUs
parameter with the change in time of the projection on the sea surface of the area en-
closed by the curve. This term gives rise to a cyclonic circulation for a decrease in the
area.

If the vertical stratification of the sea is autobarotropic (see p. 308) then N = 0 and
a change of the circulation with time can only be caused by the effect of the Earth's
rotation. If a small horizontal layer of water (area F) moves polewards, its projection
on the equatorial plane F^ will increase. If N — 0 there will be an acceleration in anti-
cyclonic circulation according to equation (X. 54). If, on the other hand, it moves to-
wards the equator it will be subject to a cyclonic circulation acceleration. The Bjerknes
circulation theorem shows clearly the importance of the baroclinic stratification of the
sea for the dynamics of ocean currents. For application see Chap. XV, 5.

(b) Vorticity for an Earth at Rest and for a Rotating Earth

A further important quantity in the dynamics of ocean currents is the vortichy.
The horizontal area F enclosed by the curve s can be divided by two arbitrary sets of
curves into a large number of very small surface elements 8F. It can readily be seen
that the sum of all the circulations SC, in the same direction along the boundaries
of these surface elements 8F, is equal to the circulation along the outer boundary s
around the entire area F.

Thus

c = £ac.

The limiting value of the ratio SC/SF is termed the vorticity and is denoted by ^.
It is thus given by

C = Hm 1^. (X.55)

The vorticity is thus the circulation around a horizontal surface unit and therefore

C = (h {u dx -}- V dy) =

idxdy = \ t 8F. (X.56)

The circulation around a closed curve s is equal to the integral of the vorticity over
the surface F enclosed by the curve s (Stokes's law). This is the two-dimensional case
and C is thus only the vertical component of the total three-dimensional vorticity
vector (curl V).

For a horizontal surface element 8j.8y (see Fig. \36d), along the boundary (in a
positive sense of rotation) of a horizontal surface element

8C — u dx -i- \v -\-
and from (X. 55)

8v
dx

\ / Su \ (8v du\

8,j - |„ + _ Syj - „ S^ = (- - -j 8.V Sy (X.57)

' 8v du\

334 Forces and their Relationship to the Structure of the Ocean

In the three-dimensional case analogously

dw 8v £u dw .. 8v du

i =

(X.59)

If the velocity has a potential (see p. 325) the vorticity will vanish and the movement is
irrotational (vorticity-free potential current).

y+by

y -

Xi-dX

Fig. \36d. Rectangular surface element for the derivation of vorticity.

The vorticity for polar co-ordinates can be derived in a similar way and it can be
assumed that the Earth and the co-ordinate system which is rigidly connected with it
rotate with constant angular velocity a>. The vorticity is then made up of the vorticity
of the rotating Earth and the relative vorticity of the water moving relative to the Earth.
To derive the vertical component Ca of the absolute vorticity it is necessary to consider
further a surface element 8F formed by the intersection of two latitude circles and two
meridians. If the latitudinal difference is d(f> and the longitudinal ^A, then the total

area SF is

8F = R^ cos cf> 8<f> SA.

The zonal velocity along a latitude circle 4> is u = RQ cos ^, where i3 = tu + dXjdt.
However, along a meridian A the meridional velocity is v = R(8<f>l8t) and some
simple calculations give for the vertical component of the absolute vorticity

S/^ 1 S2JL 1 ^

L =

8F cos </) 8X8t cos ^ 8(f)

- [Q cos2 cf>].

(X.57a)

For a small water column at rest relative to the Earth 8Xj8t = 8<f)l8t = 0, Q = oj
and the vertical component the vorticity t,E of the rotating Earth can be derived from

Ca as

^E - 2cu sin </.=/, (X.58a)

thus equal to the Coriolis parameter.

The relative vorticity c, of the water movement relative to the Earth (u zonal, positive
towards the east; v meridional, positive towards the north) is then

L'-f =

(m cos (f)).

(X.59a)

R cos ^ ^A R cos (f>

For small oceanic areas in which the latitude can be regarded as approximately
constant, equation (X. 59a) reduces to

'8v

ta-f +

8u\

8x cy'

(X.60)

Forces and their Relationship to the Structure of the Ocean

335

The vertical component of the absolute vorticity is thus always equal to the sum of
the relative vorticity (vertical component) and the Coriolis parameter,

Starting from the horizontal equations of motion (without frictional effects),
equation (X. 16) gives

du
~dt

-fv

1 dp
p dx'

8v \ cp

ot -^ p cy

(X.6])

Taking as a first approximation that p is independent of x and y or assuming baro-
tropic conditions so that p — p(p) (a function of pressure only) then, by cross-wise
differentiation of these equations and subtraction and simple calculation considering
dfjct = 0 gives

^^^^ + a+ f) divH r = 0; ia=i-\-f (X.62)

This is the relative vorticity theorem of Rossby (1939); it is used for the analysis of
stream fields in steady currents and for the analysis of moving oceanic waves.
The total change in the Coriolis parameter with time is

d4>

d<f>

-,- = Zoj COS 6 —r and smce y = — -. .
dt ^ dt R dt

The theorem of relative vorticity then takes the form

df
dt

2(jo cos (f) 2a) cos (f)

V = pv with /3 =

(X.63)

(X. 64)

R " ''^ '' ' R

If the horizontal current (m, v) is non-divergent then equation (X. 62) reduces to

i-^"-

(X.65)

The quantity /3 = cfjcy is called the '' Rossby parameter'' and represents the meridional
change in the Coriolis parameter (change with latitude). It is positive in both hemis-
pheres so that the relative vorticity always increases when small elements move
southward and decreases when they move northward.

The value of /3 at different latitudes is shown in the following Table 113.

Table 113.

10" ^ [cm-i sec-i] =

90°
00

75° ; 60' 45° [ 30°

I i

0-593 1 1145 1-619 1-983

15° 0°
2-212 2-290

In theoretical practice ^ is usually taken as a constant, that is, as independent of j\
This approximation is more or less justified near the equator where /S is a maximum

336 Forces and their Relationship to the Structure of the Ocean

and its change with latitude amounts to only a few per cent. In higher latitudes, how-
ever, taking ^ as constant is only a rough assumption since between 45° and 60°
the increase in ^ is about 29%.

If the current is non-divergent then from equation (X, 62) it follows that

|(^+/) = 0, ^« = ^+/= const. (X.66)

In a non-divergent, barotropic current the vertical component of the absolute vorticity
is constant and the change in the relative vorticity must be compensated by a corre-
sponding displacement in latitude.

To use the vorticity equation for a water mass of thickness h which is variable with
both time and space, it is necessary to take the continuity equation for the water layer
// into account in addition to (X. 62). For a horizontal current (w, v) it is easy to show
that the continuity equation must have the form

dh dhu dhv ^ dh , ^. ^ ,^^ ^_.

^+-^-f-p=0 or -w;-h divn v = 0. (X.67)

dt ex dy ot

Combined with (X. 62) this gives

It is obvious that the relative vorticity now is variable not only with latitude, but also
with the thickness of the water layer under consideration. The value it, + /)/// is thus
invariable for a given water mass; it is termed tht potential vorticity.

Chapter XI

The Ocean at Rest (Statics of the Ocean)

1. The Basic Static Equation and the Conditions for Static Equilibrium

If a water mass in the sea is at rest relative to the Earth, the only external force
acting on it will be the conservative force of gravity. In the stationary state its effect
is balanced exactly by the resistance of the masses underneath. The elastic force of the
substratum is thus opposed by the weight of the water masses and any vertical dis-
placement is extinguished, when both effects are equal (i.e. when the weight of the
water masses above any surface is equal to the pressure exerted upwards by the water
masses underneath this surface). The condition for internal equiUbrium thus requires
that no resultant of the gravity and the pressure force should act in the direction of
the gravitational level surfaces. A horizontal cross-section through a water column
enclosed between two vertical walls will carry a greater weight of water the deeper it is
placed. At a depth z it shall be p^^. At a small distance dz below this there will be a
pressure

dp
A = A + 7- ^-.

The increase in pressure p.-^^ — p^ will be identical with the weight of the water masses
per unit area between the two surfaces :

P2— Pi= pg dz.
From these two equations the "basic static equation" is obtained

1 dp

Since the negative derivative of the potential <P with respect to z is equal to the gravi-
tational acceleration, equation (XI. 1) can also be written in the form

d0 = - adp. (XI.2)

It contains the simplest statement about the three-dimensional fields of potential,
mass and pressure in hydrostatic equilibrium. The gradient of the potential is per-
pendicular to the level surfaces and the pressure gradient is vertical to the iso-
baric surfaces. Since they have opposite directions the equi-potential surfaces and the
isobaric surfaces must coincide if there is hydrostatic equilibrium. The equation
(XI. 2) states further that at any point the ratio of the thickness of a thin potential
sheet d0 to the thickness of a thin isobaric sheet dp will be constant and taken with a
negative sign must be numerically identical with the mean specific volume in this layer.
From this it follows that in the case of static equilibrium the isosteric surfaces must

z 337

338 The Ocean at Rest {Statics of the Ocean)

also coincide with the isobaric surfaces and with the surfaces of equal dynamic depth.
If the three-dimensional fields are represented by unit layers then each isobaric unit
layer is then composed of several equi-potential unit layers.

As shown on p. 308 this can also be expressed as follows: In the case of static equi-
librium there exists at the same time a state of homotropy between the three-dimen-
sional fields of mass, pressure and potential; the mass field is thus barotropic. Since
the specific volume is lawful dependent on the temperature and the salinity the state of
a basic equilibrium will also include thermotropy and halotropy.

2. Quasi-static Equilibrium and its Importance in the Dynamic Evaluation of
Oceanographic Observations

Hydrostatic equilibrium in the sea occurs only when the water masses are at com-
plete rest. If currents are present the homotropy of the three-dimensional mass, pres-
sure and potential fields will be disturbed and equation (XI. 1) is no more exactly
satisfied, since the vertical acceleration has to be taken into account in the third
equation of motion (see p. 321). However, the water movements present in the sea are
in most cases so weak and are, moreover, almost entirely horizontal, that deviations
from static equilibrium will be extremely small. This means that to a close approxi-
mation equation (XI. 1) can be regarded as valid, and it has indeed been used to
calculate the pressure field (see p. 304) from the mass field given by observation. This
fact is of very great importance in oceanography, since it permits the determination of
the geophysical oceanic structure along any vertical without a knowledge of the currents
present.

Over small areas of the sea (a few km^) the deviations from hydrostatic equilibrium
can hardly be detected. However, for larger areas of the ocean when the distance
between oceanographic stations is greater, the inclination of the surfaces of equal
specific volume relative to that of the isobaric surfaces and the inchnation of the iso-
baric surfaces relative to that of equal dynamic depth are clearly evident ; the oceanic
structure is usually baroclinic. In practice, therefore, hydrostatic equilibrium can be
assumed for each station as representative of a very small oceanic area and the
pressure field can be calculated from the mass field according to the methods already
described; however, this apparent static equilibrium changes step-wise in vertical
direction from station to station (quasi-stationary state) and the inclination of the
equi-scalar surfaces relative to each other manifests itself in this way (Fig. 137).

Rapid estimation of the relative inclinations of the isobaric surfaces in a mass field
can be made in a simple way using the equations of equi-scalar fields and the basic
hydrostatic equation (Sverdrup and co-workers, 1946). The isobars and isopycnals
in a dynamic section are defined by the equations

^cix+T dy = 0 and ^^ dx + ^^ dy = 0. (XI.3)

dx cy dx oy

The inclination of these surfaces is thus

dpjdx

dpjdx ^ dpjdx

and

The Ocean at Rest {Statics of the Ocean)

339

Fig. 137. Quasi-static equilibrium in the ocean, A and B: two oceanographic stations.
At station A the pressures Pi, p^, P3, etc., under the assumption of static equilibrium are
found at the dynamic depths Di. D^, D3, etc., on the contrary at station B at the dynamic
depths D4', D2', D3' etc. From this the inclination of the isobaric surfaces relative to that
of the equi-potential surfaces can be deduced for the oceanic space between A and B.

Taking the hydrostatic equation (XI. 1) gives after some rearrangements

e . , .dp

and from this

{ph\ — (Ph)i =

For a dynamic section the integral can be directly evaluated giving

{piX — (ph)i = h(p2 — Pi), (XI.4)

where ij, indicates the mean inchnation of the isopycnals. Introducing a mean value
of the density p in the thin layer under consideration the inclination of the upper iso-
baric surface relative to that of the lower ones is obtained

'i>i

Ipo =

. P2

approx. — ia(Si — So)

(XI.5)

if the densities are replaced by corresponding anomaUes of specific volume. This
equation permits the relative inclination of the isobaric surfaces to be readily deter-
mined from the distribution of the specific volume anomaly in a dynamic section. It
also allows a determination of how closely isobaric and isosteric profiles fit together
in dynamic profiles that have been obtained and plotted from oceanographic data.

3. Disturbances and Re-establishment of Static Equilibrium

According to the principle of Archimedes, a stationary water mass will remain
floating and at rest within a more extended water mass if its weight is equal to the
weight of the displaced water. If it is heavier than the surrounding water it will sink

340 77?^ Ocean at Rest {Statics of the Ocean)

under influence of a downward force. If it is lighter the corresponding upward force
will cause it to rise. The forces initiating vertical displacements can be easily found
from the third equation of motion in equation (X. 16). Neglecting Coriolis forces and
friction they are given by

dw dp

'dt^^~'"'8z'

If the surrounding water masses are in hydrostatic equilibrium and have a specific
volume a' then

From these two equations it follows that the enclosed water mass will be subject to
an acceleration given by

dw a' — a , ^

The upward or downward forces (buoyance force of Archimedes) is thus proportional
to the difference between the specific volumes of the surrounding and the enclosed
water masses ; for water masses of either the same sahnity and with a temperature
difference of 10°C or of equal temperature and with l%o difference in sahnity, the
magnitude of this acceleration is about 1 cm sec~^ or about one-thousandth of the
gravitational acceleration.

The nature of the equilibrium in a water column is dependent on the oceanographic
structure and is shown by the acceleration acting on a small quantum after vertical
displacement. The vertical equihbrium conditions that may occur in the ocean and
the calculation of the vertical stability that characterize these states have been discussed
in detail in Pt. I, particularly in Chap. V, 5. p. 196. It seems sufficient to refer here only
to the previous statements.

In a system where there are no forces acting other than gravitational acceleration
and the internal forces, a dynamic vertical section showing isobars and isosteres
allows an immediate estimation of the direction of the water currents produced by
the resultant forces due to density differences. Part of such a section is given in Fig.
138; the isobars can be regarded as horizontal and the inclination of the isosteres

Water of lower density

Woter of greater density

/" ..-""^ ^--""K^

^^-"""^ ---'"^ ^'-P

""" 4.--"''' ^ -"''"'' ^\

^^--'"''' ^-""""'

^.^--^"-^^ I

L -^^^ ,^- '""

' 1-'^ ' . - ' '' .--''""

'^'-"""""'^--^"""""'^-'— "-""""'"'"''

Fig. 138. Dynamic vertical cross-section: p, isobaric; a, isosteric surfaces. Disturbed
equilibrium and return to equilibrium state.

The Ocean at Rest (Statics of the Ocean) 341

relative to them show that the system is not in static equilibrium (disturbed equih-
brium). The water at A is lighter than that at B in the same isobaric level, so that to
estabhsh hydrostatic equihbrium the water mass at A must rise and that at B must
sink. The forces indicated by the mass distribution (solenoids) show a rotational
movement (circulation) which tends to adjust the mass distribution closer to that of
static equiHbrium, In the final state the isobars must run parallel to the isosteres;
a barotropic mass field is then estabhshed out from a baroclinic one. The direction
of the circulation set up is given by the rule that it always proceeds along the shortest
path from the mobihty vector B(da8n) to the pressure gradient G(8pldn) (Fig. 138).
The strength of the forces and the intensity of the resultant circulation have been dis-
cussed in II/5; see Fig. 136c. A more convenient method of characterizing the nature
of the equilibrium is by comparison of the piezotropy coefficient of the density yp with
the barotropy coefficient Fp (see p. 308). The first determines the behaviour of an
individual small element on changes in pressure (depth), while the second characterizes
the state of a water mass in vertical direction. If Fp = yp then the mass field is not
aff'ected by an interchange of any two small elements. In autobarotropism the equili-
brium condition is thus indifferent (neutral), for Fp > yp it will be stable and for Fp < yp
it will be unstable. Since in the first case the density diff'erences set up by vertical
displacements will tend to return the displaced elements to their initial positions, while
in the second, on the other hand, they will tend to displace them further and further
from it. Rhythmic (periodic) circulatory movements may be set up in this way, but in
the sea, according to their nature, they can hardly persist for very long since the
energy of these movements will soon be dissipated by turbulence (Inertia oscillations,
see Chap. XIII, 6.

Chapter XII

The Representation of Oceanic
Movements and Kinematics

1. Methods of Observation and Measurement of Oceanographic Currents

Two different methods can be used to determine the nature of the currents in the sea.
One follows the Lagrange approach and investigates the track which a small element
of water follows in time. This gives the trajectory of the water movement from the
sequence of points in space through which the water element passes. The other
method using an approach closer to that of Euler considers the current from a fixed
point, and shows the nature of the current at a fixed point at any particular moment
in terms of the current vector, which is variable with time. Graphic representation of
the distribution of velocity in space by fines of equal intensity (isotachs, velocity
field), or by representing the directional field by means o^ stream lines (see p. 326). The
stream lines and the velocity field fix the current field at any particular instant.

The trajectories and stream lines must be carefully distinguished; they will coincide
only in the case of a steady current and here the stream line will also be the same as
the trajectory taken by a small water element.

(a) Drift Bottles and Drifting Objects

A more or less accurate indication of the direction and velocity of water currents
can be obtained by following the drift of objects of all sorts which may temporarily
or permanently be floating in the water, whether through change or through having
been placed there deliberately by man (Krummel, 1908). It is essential that these
drifting bodies should project as little as possible out of the water so as to minimize
the important influence of wind and waves on their displacements.

The course followed by drifting bodies of this sort, which are subject only to the
effect of the currents, gives the trajectories of the water movement. Floating bodies
put into the sea especially for this purpose may also be used {drift bottle, bottle post).
On account of their cheapness and simple handling drift bottles have been frequently
used, and with systematic and methodical work can give useful results. Since the path
followed by a drift bottle depends to a considerable extent on chance, unambiguous
results are given only by systematic work and by the use and recovery of a large
number of such bottles. Large-scale experiments of this type have been made by
Prince Albert I of Monaco (1889) in the eastern North Atlantic, by Fulton (1897)
in the North Sea and more recently, with particular success, by Carruthers (1954)
in the southern part of the North Sea and the English Channel.

The ordinary drift bottles usually give only the starting position and the place of

342

The Representation of Oceanic Movements and Kinematics 343

recovery of the bottle ; an approximate mean value for the velocity of the current can
be calculated from the path which the bottle is presumed to have taken and the inter-
val between the two times. Large errors may occur in both these numerical values.
These circumstances have brought the method into disrepute, but as shown by the
results of Carruthers and Tait (1930) with the use of care and frequent repetition
it may still give a good idea about the system of currents over small areas of the sea. See
Thorade (1933fl) for further details.

More accurate knowledge of the course of the currents can be obtained by following
the course of the drifting body directly by means of continuous triangular measure-
ment from three fixed points. Kruger (1911) and Schulz (1925) have used this
method for the investigation of the currents in the Jade near Wangeroog and off the
Flemish coast and have obtained valuable results.

{b) Calculated Displacement

The method of determining the course of the currents at the surface of the ocean
most used in practice depends on the comparison of an astronomical position with a
position given "by dead reckoning". The first gives the true position of the ship found
by astronomical observations and the latter gives the position of the ship as calculated
from the course steered by the ship and its speed, taking the wind-drift of the vessel
into account, and the distance covered according to the log (the position by dead
reckoning). Usually this does not coincide with the astronomical position of the ship,
since it has been calculated from the apparent speed of the ship in the water. The
difference between the two positions is called the ship's displacement and is considered
to be due to currents in the time interval between successive positions (usually de-
termined at noon). For example, a ship with a noon position 52° 25' N., 42° 16' W.
(Fig. 139, point A) has travelled 225 nautical miles in the water in the direction
S. 35° W. by the following noon. The triangle AA^C gives the difference in latitude
between A and the position by dead reckoning A^, = AC = 184 nautical miles =
184 minutes of latitude. The difference in longitude A^C is 129 nautical miles. Division
by the cosine of the mean latitude gives the difference in longitude in arc minutes as
3° 24', while the difference in latitude is 3° 4'. The position by dead reckoning at
point Ao is thus: 49°2rN., 45°40'W. Astronomical observation, however, gave
49°44'N., 46°22' W. Thus

</,j = 49° 44', 9^2 = 49° 21', Acf, = 23', A^B = 23 nautical miles;

Ai = 46°22', A2 = 45°40', ZlA = 42', A5 = 42' cos 49° 32' = 27 nautical miles.

From these values the drift A^A^ is 35-6 nautical miles and y = 49° 47'; it is thus
N. 50° W., 36 nautical miles. The calculation can be considerably shortened by the
use of numerical or graphical tables.

Usually the ship displacement is regarded as the effect of an ocean current, so that
displacement = current. This is not entirely correct, since the drift includes all the
errors which have been made during the calculation of the position by dead reckoning
and during the astronomical determination of the position (see Meyer, 1923). It
can fairly safely be assumed that all the errors in both determinations are due mainly
to chance; thus the mean of a sufficiently large number of displacement values at

344

The Representation of Oceanic Movements and Kinematics

•

52°

/Aa

/^^\

51°

/ /

50°

■ 4/

- b

V'/

49°

4fl°

47°

46°

45°

44°

43°

42°

Fig. 139. Drift method for the determination of surface currents by the difference between
the astronomical position and the position according to dead reckoning. Ship's displacement :
A, position at noon of the previous day; A^, astronomical position; A^, position according
to dead reckoning; A^A^, ship's displacement; BA^, difference in latitude; BA2, difference

in longitude.

any point will therefore give the true mean current at that point. However, this of
course will only apply when the current is more or less a steady one.

Ship displacements give only the mean values of the currents over 24 h. If the in-
stantaneous value of the current or a continuous record at one position is required
then current measurements will be necessary. These will give the direction and strength
of the current, both at the surface and also in layers beneath it. For measurements of
this type at any point 3. fixed reference-position is needed. It is thus necessary to anchor
the vessel from which the observations are to be made. In shallow waters this oifers
no difficulty but at great depths the difficulties increase considerably and a special
technique and anchoring equipment are required.

If the vessel is firmly anchored and the anchor holds it is not necessarily a fixed
reference-point from which current measurements can be made directly without more
ado. Any ship anchored with a long cable will be subject to movements due to the
changes in the wind and the current, and these movements can have considerable
effect on the current measurements made from the vessel. Three types of ships'
movements can be distinguished (Defant, 1932, p. 7). The first two types, swinging
round (Schwoien) and swinging (Schwingen) are shown by changes in the heading of
the ship with time and can be determined by continual readings of the ship's compass.

The Representation of Oceanic Movements and Kinematics 345

"Swinging round" is the oscillation of the ship with the cable about a certain point
which in the extreme case will coincide with the fixed end position of the cable at the
sea bottom. Between one position of the vessel at A to another at B there will be a
change in angle y corresponding to a change in the course of the vessel from ^ to a
or vice versa. If the twisting forces of the wind and the current acting on the ship are
in equilibrium the position of the vessel will be stationary for a constant heading.
If, however, there is a change in these forces the position of the ship will alter and it
will tend towards a new equilibrium position. Thus, for example, if the wind conditions
are constant a periodic tidal current will move the vessel from a position A io B and
back again in about 6 moon hours. If the combined length of the cable and the length
of the vessel until the suspension point of the cable and the length of the vessel to the
suspension of the current meter is projected on the sea surface, then the length of this
projection is denoted by r. Since

AB = ry

180^

for r 500, 1000, 2000 m and y = 20° the velocity v of the vessel will be v = 0-8, 1-6,
3-2 cm/sec. These speeds are thus rather small provided the swinging round period is
sufficiently long and will scarcely cause errors of any importance in the current
measurement. The current meter is displaced from Aio B during such a movement and
will thus simulate a current from B io A which will be superimposed on the actual
current in current measurements, "Swinging" (Schwingen) can be regarded as an
extreme case of swinging round. The centre of the swing is shifted to the point where
the anchor cable is attached to the bow of the vessel. These oscillations will be recog-
nizable from the occurrence of a cable azimuth. In swinging movements with a period
of about an hour, the simulated current velocity will remain small also for large
values of y due to the small distance r (y = 60°, r = 60 m, r = 1-7 cm/sec), but when
the period becomes short errors will increase so strongly that the current measure-
ments will be unusable (y = 60°, r = 60 m, period = lOmin, v = 10-5 cm/sec).
However, an instrument suspended at a certain depth (such as a very long and strongly
damped pendulum) will react to the movement of the vessel and it is improbable that
it will behave very differently from the vessel. If the period of the current meter plus the
suspension wire and the swinging period of the ship are very dilTerent then the current
recorder at a deeper level will be unable to follow the movements of the vessel and
the measurements will give good results. If, however, the period of the entire system is
of the same order of magnitude as the swinging period there may be rather large dis-
placements of the current meter and the measurements will be erroneous. (Examples
are given by Defant, 1932; Defant and Schubert, 1934.)

The third type of ship movement is yawing (Gieren). The pull of the cable and the
forces acting on the vessel (wind and current) keep the ship in an equilibrium position.
If there is a change in the wind or the current the vessel will be displaced into a new
equilibrium position whereupon the cable will either tighten or slacken. Thereby,
the heading of the vessel will not change, but the angle between the cable at the bow of
the vessel and the vertical will be altered. However, at a deep anchorage the change in
this angle will be small, since the upper part of the cable will almost always be approxi-
mately vertical. The displacements of the vessel due to yawing may be considerable

346

The Representation of Oceanic Movements and Kinematics

and may be unpleasantly noticeable in the current recordings, even when the yawing
movements are unnoticed in the open ocean. Only careful determination of the
position will allow a decision to be made regarding the extent to which the move-
ment of the vessel due to yawing has affected the recordings.

An excellent example of yawing movements was obtained at the "Meteor" anchor station 228 in
the strong North Equatorial Current and the intensive north-east trade-wind. Figure 140 shows that
there was a "freedom in the yawing movement" of 2473 m and the mean position of the ship was at a
distance of 2760 m from the anchor point when projected on the sea surface. Since while the vessel
was anchored, the direction of the wind was hardly varied and since there was a strong basic
current (22 cm/sec), and a wind force of 5-6 Beaufort, the weak tidal currents could hardly move the
vessel from its main position and the movement of the vessel must have been due to yawing. Actually
as shown in Fig. 141 which gives the positions of the vessel determined astronomically while at
anchor, there were almost only yawing movements. It can be seen that all factors (heading of the
vessel, position, etc.) fit excellently to give a very plausible representation of the movements of the
vessel while anchored (see also Defant, 1940«).

iooo4

2000

30004

4000 -r

4486^^^^

Fig. 140. Anchor station 288 of the "Meteor": water depth 4486 m, length of cable 6003 m,
freedom for yawing 2473 m, anchor position — mean position of the ship 2760 m.

Since a really fixed point is scarcely obtainable in the open ocean at great depths,
methods have been devised for the elimination of errors that occur for this reason in
current measurements. Witting (1930); Thorade (I933fl) have reviewed the three
methods so far used to replace the absolute method using a fixed point.

The correction method consists essentially of a careful observation of changes in the
position of the ship relative to that of a buoy anchored by the shortest possible cable
and in the correction of the current recordings by use of these observations. Witting
(1905) has given a procedure for calculation using numerical and graphical methods,
but due to the complexity of the observations and the difficulty of evaluation of the
measurements it has seldom been used. In the difference method the vessel is not an-
chored to the bottom but is kept stationary with a driving anchor; the current re-
corder then gives only the movements of the water relative to the ship. To find the
true current it is necessary to know the absolute current at one of the depths investi-
gated. Hansen (1915) and later Helland-Hansen (1926), for want of other possi-
bilities, used a second current recorder close to the sea bottom, or as deep as possible,
and assumed that the water here would be almost motionless. For current measure-
ments in the ice drift off the North Siberian Shelf Sverdrup (1929) on the "Maud"

The Representation of Oceanic Movements and Kinematics

347

47»3a'

47''37'

47« 36'W

47«>35'

l2°3gN

2138' N

I2»37'

47" 38'

4r3r

47''36 W

47° 35'

Fig. 141. Successive positions of the ship, ship's course and circle of yaw at the anchor
station 288 of the "Meteor", 27-29 March 1927.

ip's positions:

27.iii.

21.45 MGZ

12=

37-8' N.

; 47' 35-2' W.

28.iii.

08.54 MGZ

J20

37-4' N.

; 47° (36-7' W.)t

28.iii.

12.00 MGZ

12^

38-0' N.

47° 35-9' W.

28.iii.

21.47 MGZ

12"

38-0' N.

47° 35-6' W.

29.iii.

08.55 MGZ

12 =

37-5' N.

47° 36-8' W.

+ The computed longitude of 47° 37-7' W. is very probably an error; 36-7' W. should be
the correct value.

modilied the method by using a sounding line to obtain a fixed point at the sea bottom,
but this can only be used in very shallow waters.

According to Witting, the best, fastest and also the most frequently used method is
the "'smoothing method''. The current measurements are made from an anchored vessel
at the shortest possible intervals and values for a time interval over which the different
movements of the ship almost cancel out are combined to give a mean vahie. An
interval of about 1 5-30 min seems to be sufficient to eliminate the variations due to
the movements of the ship and irregular changes in the current direction and speed.

(d) The Scientific Use of Current Measurements

The technical refinements of the operative mechanism of the amazingly large
number of current recorders used in oceanography need not be discussed here;
reference can be made to Thorade (1938^), Sverdrup and co-workers (1946) and
particularly to Oceanographic Instrumentation Isaacs and Iselin, 1952). However,
the important subject of the scientific use of current measurements will be dealt with
here in greater detail.

348

The Representation of Oceanic Movements and Kinematics

The individual values obtained from current measurements as discussed above will
contain errors due to the simultaneous movement of the vessel, and correction to the
true current can only be made if the movement of the ship is known with some
accuracy. Since for current measurements in the open ocean only one current meter
records on board ship, the correction method of determining the true current cannot
usually be used. If the average true current changes only slowly, the smoothing method
of ehminating short period movements of the vessel must be appHed. How strongly
the observations have to be smoothed has been shown by Thorade (1934) with
observations made by the research vessel "Poseidon" in the Kattegat (August, 1931).
The Rauschelbach current meter was used here to give continuous records of the
current every 10 sec over a long period. Plotting all these current vectors starting from
a single zero point of an appropriate co-ordinate system gives a current diagram of
the type shown in Fig. 142. The individual current vectors are strongly scattering and

Fig. 142. Recordings of the Rauschelbach current meter at the anchor station of the
"Poseidon" in the southern Kattegat during I h for each 10 sec. (10 August 1931; 18.30-
19.30 h). The current arrows must be drawn from the point O towards the crosses. The
indicated arrow refers to the start of the observations. The dotted line shows the movement
of the arrowhead during the following 3 min. The dashed-dotted line indicates the position
of the arrowhead after smoothing, the point O shows the mean position during the h h.

their end-points form a point cloud covering a relatively large area. It can hardly be
assumed by the values given in the diagram that the true current has altered significantly
within the half-hour observational time. The dashed line joins the end-points of the
vectors for the first three minutes. Even for this short interval the vectors cover almost
the entire area of the point cloud. This shows that single current measurements made
from an anchored vessel differing widely in the observation time are more or less worth-
less. It is rather different, however, if for short observation intervals mean values are
taken for more or less long intervals in time. Fig. 143 shows that for the same values
as in Fig. 142 the individual means for each minute are rather scattered, but the means
for intervals of 5 min, on the other hand, show only small variations during the half

The Representation of Oceanic Movements and Kinematics

349

hour. These findings by Thorade indicate that the effect of the movements of the
vessel from which the measurements are made and other chance factors can be eUmin-
ated by such a smoothing procedure. Instead of using continuous recordings of the
current followed by calculation of the mean over a long interval such equipment is
used in practice which gives directly mean values for the direction and velocity over

R -20

■30L

,830 1835 1840 1845 1850 |855 19OO
IO-2ni-l93l

-10

,E, -20

^"^^"^

+ ^'

.-"^

— ?~--^— -^

1830 le

35 16

40 ij

j-^s le

50 18

55 19C

IO-5fflI-l93l

Fig. 143. Upper picture: mean for each minute; lower picture: mean for each 5 min of the

north ( \ \ ) and the east component (— O — O— ) of the current measured from

the "Poseidon" (see Fig. 142).

a longer interval (10 min or more). In deriving the means it should be remembered
that they are vectors and in order to reduce them to mean values they must be re-
solved into north and east components. The mean obtained in this way is denoted
the vectorial mean. Instead of this mean, which is mathematically accurate but in-
convenient to calculate, the mean of all the velocities regardless of the direction is
often used instead. This is termed a scalar mean of the velocity, and it represents the
average velocity of the water displacement. The corresponding simple arithmetic
mean of the angle of the flow direction is of no importance especially when the
variations in the direction are large.

In the characterization of extensive current measurements a further quantity is
used to give a numerical value for the variations in direction and speed of the current.
The quotient of the vectorial velocity mean and the scalar mean is used for this and is
termed the constancy (stability) of the current {Kgl Ned. Med. Inst. De Bilt, 1904,
1908). From the definition of the two kinds of averages it follows that the stabiHty
is always a proper fraction. It has the value 1 if the directions of the individual vectors
are always the same size, since the vectorial mean is then the same as the scalar. The
current constancy is usually expressed as a percentage.

350 The Representation of Oceanic Movements and Kinematics

The magnitude of the current constancy is only affected to any large extent by varia-
tions in the direction of the flow, variations in the velocity have little influence.
Wagner (1932) has found, for example, that if the velocity was assumed to be the
same for the individual values and the directions were scattered within an angle of
90°, then the stability was 90-100%, while if the directions were scattered within 180°
the current stability was still 60-90% ; individual values with greater velocities could,
however, aff'ect these stability values strongly in either direction.

A more accurate description of the distribution of a larger number of obser\'ations
requires the use of statistical theory (Thorade, 1936). If the measured velocities of
the current are m\, w^, Wg, . . ., vv„ for ^-observations and a^, og, a^, . . ., a„ are the
corresponding directions (taken clockwise from north from 0° to 360°) then the
corresponding ^-components will be m^ = m,\ sin a^ and the A^-components will be
Vi = Wi cos ttj, where / = 1, 2, . . ., n. The arithmetic mean of the ^-components
will be a, and that of the A^-components v ; then the vectorical velocity is

w^^ = u} + y2
and the vectorial mean direction will be

u
tan a„ = -.

The deviations of the individual values from the vecto^'ial mean are

^i — Ui — u and t^j- = f, — v.

Comparison of the frequency distribution with a Gaussian distribution will then allow
us to judge whether the deviations are generally random, so that statistical laws are
applicable.

The mean scatter of the Mj- and ^j-values is then given by the mean error (standard
deviation)

w,/ = e* and m^ = if.

For a case similar to that of Fig. 142 (150 observations over an interval of 10 sec)
Thorade found a point distribution given in Fig. 144 for the frequencies of the devia-
tions for intervals of 1 cm/sec; the curves show a Gaussian distribution indicating
the completely random nature of the deviations, and show that in spite of the small
number of observations the deviations approximate very closely to a random distribu-
tion. In this way, the direction varies between 270° and 318° and the velocities between
7-4 cm/sec and 21 -8 cm/sec. The vectorial mean gave a current N. 66° W., 14-5 cm/sec,
the scalar mean was 14-7 cm/sec, and the current constancy (stability) was therefore
98-6%; m spite of the rather large variations in direction and speed of the current
this is a surprisingly high current stability value. The mean scatter gave a considerably
better idea of these variations: m„ = ± 2-68, m„ = ±2-64 cm/sec, which indicates
that for a random distribution of the deviations about 68% of all the deviations ej
ofthe£'-component lie between +2-68 cm/sec and —2-68 cm/sec; analogous conditions
apply for the 77^ for the A^-component.

According to statistical theory of scattering, the direction and velocity can be charac-
terized most accurately by the "mean error ellipse" which must include half of all
the individual values. Considerable numerical work is required for calculating this

The Representation of Oceanic Movements and Kinematics

351

-8

-6 -4
cm /sec

-2
/ +

3 +

+
~20N^

-15

2
\

4 6

cm/sec

East
component

-10
5

+ +

-6 -4

cm/sec

4 6

cm/sec

Fig. 144. Frequency distribution of scattering of the north and east component e and -q for
the point cloud of the current measurements of Fig. 142 (the full lines indicate the Gaussian

frequency distribution).

ellipse. In place of it Thorade used the scatter circle the radius of which is given simply
by p^ = m^^ + m^. This circle is quite sufficient for the characterization of the scatter
of a point cloud of current values. The probabihty that an observation will fall within
the scatter circle is with sufficient accuracy about 2/3, that is, about 2/3 of all observed
values will fall within the scatter circle. In the case previously mentioned (see Fig.
143). p = 3-76 cm/sec; the actual number falUng within the scatter circle is 103 of
the 150 values, which is about 2/3.

Elimination of periodic components. The variations in speed and direction of ocean
currents often include periodic components superimposed on the mean current {the
basic current). The basic current because it is often obtained by elimination of the
periodic components is therefore sometimes rather unsuitably called "residual current".

The basic current need usually not to be constant either in direction or velocity, but
these changes are mostly aperiodic and of long duration and therefore differ consider-
ably from the periodic components. The presence of these components is shown par-
ticularly well by graphical representation of the individual vectors in a progressive
vector diagram. A constant basic current plotted in this way will give a straight line,
while a wavy or spiral trajectory indicates the presence of periodic components.
Figure 145 shows a case of this type. Generally a water transport occurs directed to-
wards west-south-west, but it is not uniform and shows wavy fluctuations to the north
and the south (period of these oscillations about 14-15 h).

The periodic components can be eliminated by taking a mean over the periods
present; the periodic components then cancel out giving the average basic current.
Thus, the case of Fig. 145 gives a mean displacement over the entire period of 2-0
nautical miles towards the south and 7-5 nautical miles to the west in 24 h or a basic
current of W. 15° S., 16-7 cm/sec.

The calculation process for such a separation of observed current values, taken over
a long interval into the basic current, and the periodic components can be illustrated

352

The Representation of Oceanic Movements and Kinematics

n 4"

■^

. 0"

2l4

18"

•;

1921

1 2
o

2053/

^ .

i£^'

/ '"

10"

14"

1 1

r:V%

y3"

^4^
'V

1
Nautical

miles

-.1 —

,. 1

Fig. 145. Anchor station of the "Ahair": path of a water particle from 19 June 1938 00.00
to 20 June 1938 14.14 MGZ. (Represented by a successive plotting of the observed current
valuesf as a mean between the depth 5 and 15 m). Mean basic current from 19.00 to 20.00
MGZ: north component: —4-5; east component: —161 = W. 15° S., 16-7 cm/sec.

t In order to simplify the numbers indicated at each individual point are values rounded
off to total hours. Therefore « h is always (« — l)h 48 m, for example, 3 h = 2 h 48 m.

by an example. The method given below is mostly used but each case requires indi-
vidual treatment. At the "Meteor" anchor station, 16-20 June 1938 (44° 33' N.,
35° 58' W., mean depth about 1400 m) the current values were measured for a single
interval of 10 min in each hour at eight depths (Defant, 1940Z)). Figure 146 contains
unsmoothed values for the N- and jp-components at 5 and 15 m depth for the period
from 17 June, 04.00 h to 19 June, 18.00 h (MGZ).

This gave a rather irregular, jagged curve, partly because of chance disturbances
and partly because of errors in the measurement. Since the tidal currents were ex-
pected to be rather strong these were then eliminated by taking continuous means
over 24 lunar hours (from one moon culmination to the next). The smoothing showed

Tim*-

I7 3ZI

4 6 8 10

Fig. 146. Current components at the anchor station of the "Altair" June 1938 (,4> = 44°
33' N., A = 33° 58' W.). — i — i — i — i — , N. and E. components according to the observation;
--0 — o--, basic current after elimination of the periodic parts (tidal current and inertia

current); 1 1 , basic current + tidal current of the diurnal and semi-diurnal wave

+ current of the inertia wave (according to the values of the harmonic analysis).

The Representation of Oceanic Movements and Kinematics 353

in the present case that the remaining current was indeed very regular but still included
a weak periodic disturbance of about 1 7 h. Since it was not improbable that a wave
of this type could occur in such current measurements (inertia oscillation) this wave
was also eliminated by taking means again over a 17 h period.* Finally, the basic
current remains. It has been plotted in Fig. 146 for both components. It changes only
slightly with time; the A^-component gradually decreases from 10 to about —4 cm/sec
and then remains almost constant, the ^'-component changes from —12 to —17
cm/sec.

A more detailed analysis of the periodic components can be made by ordinary
harmonic analysis and gives the following equations {t in hours) :

7.TT Iv

TV-component: +6-6 cos .- (/ — 17-6 h) + 4-6 cos j^{t — 2-3 h)

277 ,

+ 6-0 cos yj(t - 12-6h).
S'-component: +2-7 cos i^{t - 20-4 h) + 3-8 cos -r^ (^ - 5-2h)

l-rr

+ 5-3 cos ynit - 0-Oh).

The time ? = 0 corresponds thereby rather accurately to 3 moon hours before the
moon passes the meridian at Greenwich (17 June, 1938). The ampUtudes are given in
cm/sec. All three waves show almost the same amplitude; the inertia wave also is
quite pronounced and, as can be expected from this, can become quite visible in the
current. Calculations of the current from both components obtained by the harmonic
analysis, and in addition the basic current of the curves presented in Fig. 146, follow
the observed values very satisfactorily. However, the differences between the smoothed
curves and the observations show that the current measurement is subject to manifold
disturbances which are very largely random (or observational errors).

From the smoothed mean values for a full period of the single waves current
diagrams can be constructed and can be compared with the current ellipses which
were calculated from the harmonic values. The left-hand side of Fig. 147 shows this
comparison for the semidiurnal tide and on the right-hand side for the 17 h inertia
wave. The smoothing of the subsequent values by harmonic analysis is rather obvious

* For a curve y formed by the superposition of two harmonic waves of different periods T^ and
T2 which has the form

y = acos"^ U - ej) + 6 cos -^ (/ - eg)
^1 -'2

if a continuous mean is taken over the period T^ the Tg-wave will disappear completely and there will
be left

y = ^\ y dt = a —^ sm -;^^ cos ~ (/ - ^i).

The amplitude is changed, but not the period and the phase of the T^ wave. If Tj is 17 h and T^ is
24 h then the amplitude of the 17 h wave which was previously a will now be —Q-lla, that is, the
Tj-wave is now inverse to the original wave and its amplitude is almost five times less than before.

354

The Representation of Oceanic Movements nad Kinematics

in both cases. Reference should be made to Vol. II for further discussion of these
current diagrams. It may be mentioned here that the components, phases and ampli-
tudes of the inertia waves correspond closely to those given by the theory of oscilla-
tions (see Chap. XIII, 6). Since the current diagram deviates only slightly from a circle
the fine dashed circle in Fig. 147); for this the amplitudes of both components must be

Fig. 147. Current measurements at the anchor station of the "Altair", 16-20 June 1938.
Left side: current card of the semi-diurnal tide according to the smoothed values of the
individual hours and the current ellipse according to the values of the harmonic analysis.
Right side: the same for the 17-hourly inertia wave (the dashed circle indicates the theoretical

inertia circle).

the same, and furthermore, the phase of the ^-component must lag one-quarter of a
period behind that of the iV-component. The observations give an amplitude ratio of
M4 and 12-6 h + 4-25 h = 16-85 h as compared with 17-0 h which is a difference of
only 0-15 h. These properties of the 17 h wave confirm that it is a pure inertia wave.
Decomposition of current data by means of other methods. After the elimination of
the periodic components there still remain other more aperiodic effects superimposed
on the basic current, which is almost constant in time. These deviations may be due
to various causes such as piling up of water at shores (Anstau) or variable wind stress.
The wind especially is liable to give rise to drift currents in the surface layers, the
direction and strength of which depend on that of the wind and change with it. The
observed current in these cases can be looked upon as the resultant of thedrift current
and the basic current. If the latter alone is required the two must be separated by a
special procedure. Nansen (1902) gave a suitable method for this which was used in
the evaluation of the ice-drift observations of the "Fram". If intervals of time, for
which the wind resultant is zero, are taken together and the effect of the wind on the
water therefore considered very small, then the resultant current for the total interval
will be due to the basic current alone. In that way he found by analysis of six rather

The Representation of Oceanic Movements and Kinematics

355

typical cases that the permanent current of the deep North Polar Basin flows first
at 1-0 cm/sec N. 64° W. and later at 2-1 cm/sec S. 12° W. Brennecke (1921) and
SvERDRUP (1928, \93>\b) later used the same method to show from the ice-drift
observations of the "Deutschland" and the "Maud" that there is no permanent surface
current in either the Weddel Sea or off the North Siberian Shelf.

Later, Sverdrup developed another method that makes use of all the available wind
and current observations. The vectorial resultant of the current is calculated for wind
groups concerning certain directions (for example, four groups with the wind for each
quadrant centred on N., E., S. and W.) and divided by the wind strength of each
group to give the "relative" resultant current (for 1 m/sec wind). If a pure drift current
is present then the resultant of the current vectors of all the wind groups must vanish,
since they will be symmetrically grouped around the zero point. If, however, the ob-
served current is made up of wind drift + basic current, the resultant of all the groups
will not be zero but will represent the basic current. If a coasthne impedes the de-
velopment of the wind drift equally in all directions and favours a current parallel
to the coast, then the circle connecting the ends of all the current vectors will be
replaced by an ellipse (Witting, 1909).

Table 114. 'Tram" Expedition: 27 May 1895 - 27 June 1896. Ice drift
grouped according to the directions of wind resultants

Wind

Total drift 1 .^ Wind
1 (without ba

drift
sic current)

quadrant
centred at

Wind

speed

V (m/sec)

Current Relative
intensity current
w (cm/sec) wjv

Deflection
angle

' Relative
current

Deflection
angle

N.
E.
S.
W.

3-30
2-86
2-48
2-56

416 j 1-26 9-5°
316 ! 110 53-5°
5-54 2-23 420°
5-92 2-31 200°

1-69
1-65
1-68
1-65

360°

27-5°
24-5°
340°

Mean 2-80 _ i _ _

1-67

30-5°

Table 114 contains the ice-drift observations of the "Fram" for the period from 27
May, 1895 to 27 June, 1896 (Fig. 148) according to Sverdrup. The diagram on the
left of Fig. 148 shows that the end-points of the vectors lie on a circle, but that the
centre of the circle is not at the zero point but is displaced in the direction S. 82° W.
Vectorial subtraction of the basic current (0-79 cm/sec, bearing 262°) results in the
diagram given on the right of Fig. 148. The velocity 0-79 cm/sec refers to a wind speed
of 1 m/sec. During the year, however, the mean wind speed was 2-80 m/sec, so that
for the period under consideration there was a permanent surface current of 2-
cm/sec along a bearing of 262° (direction relative to the 75° E. meridian). Nansen
obtained by his method 2-0 cm/sec on a bearing of 256° which is in satisfactory agree-
ment. The table shows that the relative wind drift is practically independent on the
direction of the wind ; the mean of the four groups shows that a wind with a strength
of 1 m/sec gives rise to a surface drift of 1-67 cm/sec deflected 30-5° to the right of

356

The Representation of Oceanic Movements and Kinematics

From, May 27 1895- June 26. 1896

Fig. 148. Dependence of the ice drift on the wind direction according to the observations
of the "Fram" expedition, 27 May 1895 to 27 June 1896. Left side: the observed total ice drift.
Right side: the pure wind drift after subtraction of the effect of the permanent basic current

(according to Sverdrup).

the wind direction. Also in this case almost identical values were obtained by Nansen's
method.

Palmen (1930/)) has studied these methods in his work on the currents of the Gulf
of Bothnia and the Gulf of Finland more deeply and has used them with success,
especially for the observations on wind and currents made at the light ship "Finn-
grundet" from 1923 to 1927.

2. The Current Field and its Representation

{a) Representation of Mean Current Conditions by Means of Compass Cards

To get an idea of the currents in any particular area of the sea the most practical
procedure is to tabulate all the available data for the direction and strength of the
currents for small areas over which uniform conditions can be expected. These small
areas are usually chosen to cover a few degree squares (one, two or more degree
squares). The question is thus to count out a large number of observations which
can then be presented on a compass card. The prevalence of each direction is then
shown by longer or shorter rays from the centre point, and the mean velocity in any
direction is shown either by the thickness of this line or by the feathering on these
rays. Such a current chart is actually only a graphical tabulation and is very largely
free of subjective influences. A personal factor becomes involved only in the interpre-
tation of the picture shown by such compass cards.

The representation of current conditions by compass cards best satisfies the require-
ments of a current chart for navigation, since it gives at a single glance the frequency
and strength of currents in each direction and the possibility of representing large
variations in the direction and strength of the current. The usefulness of charts con-
taining compass cards for scientific investigation of the sea is, however, very limited,
because sufliicient observations are available only along shipping routes and there are
larger areas of the sea for which cards cannot be constructed due to missing data.
The use of compass cards to show average current conditions was previously pre-
ferred, and by this a uniform evaluation of the enormous amount of ships reckoning
displacements was made. One of the most recent representations using compass
cards is that of the Netherlands Atlas for East Asian waters {Kgl. Ned. Met. Inst.

The Representation of Oceanic Movements and Kinematics 357

De Bilt, 1935-6). From this atlas the part contained in Fig. 149 was taken; for an
explanation of this picture see the legend underneath.

A picture of current conditions easier to interpret can be obtained if only a selection
of the particularly typical vectors are given as, for example, in the Deutsche Seewarte
Atlas containing twelve monthly charts; however, in these the subjective viewpoint of
the investigator has a large effect. A different type of representation has been used in
the British Admiralty charts. The ship reckoning displacement is not shown by a
straight arrow, but by a wave-like arrow with the mean velocity in nautical miles per
day indicated by a number underneath. Where there is no displacement the chart is
left blank but along the usual shipping routes they accumulate. In practice this method
has the advantage that it shows the variations and the uncertainty in the occurrence
of the ocean currents and the greater or lesser prevalence of current free regions or only
of weak currents.

(b) Representation of Average Current Conditions by Means of Stream Lines

Instead of giving statistics of individual ship displacements in form of compass cards,
these statistics can also be used to give the mean value of the currents in the degree
squares. This has been done by the Netherlands Meteorological Institute (1908, 1915,
1919). A vectorial mean for one or two degree squares is taken of ship displacements,
and calculations are also made of the scalar means and the stability. The results have
been published in tables and charts. This observational material has then formed the
basis for a whole series of investigations on ocean currents. Attempts to derive a
comprehensive picture of the currents from these mean current vectors are of two types
(Schumacher, 1922); one of these represents the current by stream lines broken up
into arrows with the feathering or the thickness of the arrows indicating the velocity.
The other gives the direction of the current by continuous stream lines and the velocity
by isolines (isotachs). To the first group belong the investigations of Michaelis
(1923) and Willimzik (1927) on the Indian Ocean, of Meyer (1923) on the Atlantic, a
study by Merz (1929) on the Pacific Ocean, and by Willimzik (1929) on the Antarctic
surface current and others. The second method was first used in oceanography by
Bjerknes and co-workers (1913) for the currents in the Gulf of Mexico. During a
renewal of the monthly current charts for the North Atlantic Schumacher (1940)
later used another method of representation. The arrows here were drawn to represent
not the mean direction and velocity but the most frequent, which is more valuable both
for the practical user and in most cases also for scientific purposes. All the available
data on observed ship displacement were evaluated on this most frequent value (mode)
principle. The quadrant containing the largest number of observations was found for
each point; the enormous amount of work required was handled by a punched-card
system (Hollerith). The direction separating this quadrant into two halves was then
taken as the prevailing direction of the current. The velocity was taken as that usually
found in the prevailing direction, that is, the scalar mean of the ship displacements
falling within the selected quadrant.

Also, the stabiHty was determined as before and was characterized by the probability
of a displacement in the selected quadrant, i.e. by the numerical ratio of the number
of observations falling within the quadrant to the total number of observations. Four
different grades of stability were distinguished. If at least one-third of all observations

358

The Representation of Oceanic Movements and Kinematics

The Representation of Oceanic Movements and Kinematics 359

Explanation (to Fig. 149)
The current roses are drawn from observations within the areas shown by the pecked Hnes.
Arrows indicate direction of current; north arrow current towards N. Velocity of current
in nautical miles per day is represented as follows : e-iz '^-^-^ ?5-4b^ 49-72 TSon^ove _ Length
of arrows represents frequency, 1 mm 3-7%: j j j i [ . The lower

o 50 I °° %

figure within the circle gives the total number of observations, the upper figure the per-
centage frequency of currents less than 6 miles per day.

falls within a quadrant this will be already predominant and its middle line can be
regarded as the direction of the prevailing current. If the percentage of the ship dis-
placements falling within the quadrant is between 33% and 66% then the prevaiUng
current is termed ''variable''. The next grade ''rather steady is reached when at least
33% of all observations fall not only within one quadrant but within one octant. If
more than 61% of observations fall within a quadrant and between 33% and 66%
within an octant within the quadrant then the prevailing current is denoted "steady \
if both quadrant and octant contain more than 67% of all observations the current is
""'very steady"'. This characterization of stabihty is undoubtedly more illustrative
than the ratio of the vectorial and scalar sums of the velocities.

An example of this type of representation is given in Fig. 1 50 which shows the chart
for August of the surface currents in the North Atlantic as given by Schumacher. The
length of the arrows indicating the prevailing direction has no significance here. The
velocity is given by feathering or for large values by barbs at the arrow-heads; for
the grade of the stability see the explanation on the chart.

A similar evaluation of ship displacements has also been given by Schumacher
(1943) for the South Atlantic so that modem monthly charts are now available for the
whole of the Atlantic Ocean.

Certain definite properties of the current field must be borne in mind in plotting
stream lines on the basis of the current vectors. In the j&rst place it should be noted
that except at singular points and lines :

(1) the individual stream lines are not allowed to intersect;

(2) the stream lines are curves that neither start nor finish in the current field ;

(3) the stream lines are always continuously curved lines.

The stream lines are drawn mostly by vectorial interpolation by the eye. Such a
graphical interpolation usually offers little difficulty if the current vectors cover the
whole chart uniformly. However, this is usually not the case and the lines must some-
times be drawn with a minimum of observational values. For this it is necessary to
have some idea of the singularities in the current field (Bjerknes and co-workers 1912,
1913). Because the position of these singularities fixes the general outline of the field
and to complete the pattern then offers little difficulty.

The simplest singularities and their relationship to the structure of the water masses
in the oceans will be described in the following section.

Lines of convergence a?id divergence. Figure 1 5 1 shows convergence and divergence
from only one and from both sides of the stream lines. In case (a) and {b) there is
an infinitely rapid convergence and divergence; cases which are rarely found in this
extreme form. An infinite number of stream lines leaves or enters asymptotically

360

The Representation of Oceanic Movements and Kinematics

361

-Fig. 150. Chart of surface currents for August in the North Atlantic Ocean (according to
Schumacher). (Stereographic azimuthal projection accurate at the equator, scale at 0^ N.,
30°W. 1: 108.)

Velocity
3-0- 8-9 sm/Etm.
90-14-9

15 0-20-9

21 •0-29-9

300-41 -9

420-53-9

540-65-9

66-0-77-9

( i knots)
( 2 knots)
( f knots)
(1 knots)
(U knots)
(2 knots)
(2i knots)
(3 knots)

— <
— <-

A-

Steadiness

Very steady

Steady

Rather steady

Variable
rDead reckoning
J or taken from
1 other represent-
Lations

Numerical limits for steadiness: from all ship displacements known in a certain area fall
inside the quadrant which is cut by the current direction into two halves (quadrant and

octant, respectively)

Very steady
Steady

Rather steady
Variable

quadrant (%)
(up to 45^ to the
right and left)
More than 67
More than 67

33-66
More than 33

octant (%)

(at the most 22^° to

the right and left)

More than 67

33-66

Less than 33

Fig. 151. Singularities in the current field: (a) one-sided convergence, ib) one-sided diver-
gence, (c) and id) double-sided convergence and divergence ; for explanation see vertical

cross-section.

362

The Representation of Oceanic Movements and Kinematics

from both sides the hnes of divergence and convergence (case (c) and {d)). Lines of
convergence and divergence in most cases represent the boundaries between different
water types moving relative to each other. They are generated when heavy water meets
lighter water or when lighter water spreads out over heavier water that is sinking.
Fig. 151 gives a vertical section showing current conditions on both sides of an inchned
gliding surface separating two different water masses. Similar vertical displacements
can also be expected for divergence and convergence lines from both sides. In all these
cases where there is a velocity component at right angles to the boundary surface the
inclined gliding boundary surface cannot be expected to remain stationary.

The occurrence of divergence and convergence lines in oceanic current systems is a
general phenomenon closely connected with the oceanic circulation. They represent
the framework of the circulation and indicate the connecting places between the sur-
face currents and the three-dimensional vertical circulation. Some examples will be
given later.

Rauschelbach ("1931) while making current measurements in the Ost-Friesband Gatje rtielow
Emden) took the opportunity to make measurements with a bifilar current meter at a convergence line
running through the observation point (an anchored vessel). The convergence line, which was visible
as a foam line, ran parallel to a dredging line; it moved the Ems upstream driving with the flood tide,
while at the same time it was displaced from the middle of the channel towards the east. It passed the
current meter at 1 7 h 3 min 30 sec. Figure 1 52 gives the velocity and direction of the current as measured
by the current meter before and after the passage of the convergence line; Fig. 153 shows the distribu-
tion of the surface current around it. The course of the boundary surface in the lower layers was not
that simple and according to current measurements at a depth of 1-2 m was disturbed by internal
waves.

40
20
y I80°0
" 160
140
120

100

(a)

■^

-s/

«o-cr^*^

(b)

^'^S/Su

y^^

fsJ]

rsr

l7hQ'" |r

3m ^r

Time

Fig. 1 52. Evaluation of a convergence line in the Ostfriesland Gatje (downstream of Emden)
according to Rauschelbach: (a) current velocities, (b) current directions (clockwise from

0° to 360°).

Convergence lines are frequently indicated at the sea siirface by more or less strong
agitation of the water and are then recorded in ships' logs as rips. Closer attention has
only been paid to them in more recent times. (Romer, 1935, 1936; Schumacher, 1935;
Thiel, 1937; Uda, 1936, 1938). It seems to be definitely established that rips in the
open ocean are formed at the boundaries between converging and diverging water
masses. Sometimes when lighter and heavier water are separated by either a converg-
ence or a divergence line, the wind forces the lighter one to move above the heavier,
as is often observed. Off the continental shelf and around island platforms there may

The Representation of Oceanic Movements and Kinematics

363

also be disturbances of the water movement due to the bottom configuration; the
direction of the rips then usually corresponds with the main course of the shelf or of
the irregularity in the bottom. In many cases a connection has been shown with the
behaviour of the tidal currents in neighbouring oceanic regions. Particularly well
known to seamen are the rips in the Straits of Gibraltar and in the Straits of Messina,

-1 —
\\ \

w \
\\ \
\\\

\ \N

"*> ">
\\\
\\\

W \

\\\
\ \ \
\\\
\ \\

^^^
^^ \
\ \ \
\ \\
\\\
\\\
\ \ \

\N\
\ \ \
\ \ \
V^ \
\ \ \

\ \ \ \ \ \ I
\\\\ \ \ I
\ \ \ V \ V I

\ \\\\ \ I

W \\\ \ I

\ \ \\\ \ I
\\ \\\ \ I
w \ \\ \ I
\\ \ \\ \\
\\\\\\ I

\\\V \ \ I

\ \ \ \ \ \ I
\ \\ \ \ \ I
\ \\ w \ I

\ \V \\ \ I

\\\\ \ \ I
\\\ \ \ \ I

\ \ \ \ \ V I
\ < \ \ > V 1

\ \\\\ \ I
\ \ \ \ \ \ I

\\\\\ V I
\ \\ \^ V I

\ \ \ \\ \ I

\ N \ \ \ \ I

— I —
1 1 1 1 .

,111
i II I
1 1 1 1
till

1 1 1 1
1 1 1 1
1 1 1 1
,111
1 1 1 1

1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1

'''''/l

1 1 1 1
,111
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
till
1 1 1 1

H I I I I

mill
, 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1

1 1 1 'I I
1 1 1 1 1 1
1 1,' 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 / /
1 1 1 1 1 1
1 1 1 / I I

(0)

Fig. 153. Current directions and stream lines during the passage through the convergence
line in the Ostfriesland Gatje (see Fig. 152).

where they are definitely connected with tidal currents carrying different water types
and those off the eastern coast of North America in the area of the Gulf Stream, along
the west coast of Mexico and in West African waters where they are related to up-
welling phenomena.

Points of convergence and divergence (Fig. \54a, b). These points represent the inter-
section of an infinite number of stream lines. For continuity reasons, movements
such as these must always be connected with movements perpendicular to the surface
of the sea ; thus a divergence point in a water layer near to the surface indicates up-
welling and a convergence point indicates a sinking movement. This need not, however,
be the case at greater distances from the surface. The divergence then merely indicates
that due to the vertical movement more flows in at one side than leaves from the other;
the reverse applies for convergence. The formation of curved stream lines near to the
centre point (cyclonic and anti-cyclonic vortices) as shown in Fig. 154 depends largely
on the effect of the Earth rotation.

If there are different types of water masses in the near vicinity of the vortex they will be
drawn into it and combined singularities then occur. Cases of this type are shown in
Fig. 154c", d; c represents a cyclonic vortex in the region between a lighter and a heavier
water mass. Since the equilibrium state is upset at the boundary between the two water
masses the hghter water tends to spread out over the heavier while the heavier sinks
underneath the lighter. For such an inward spiraling motion a convergence line forms
at the boundary surface ; thereby one part of it will be an up-gliding surface where
the hghter water moves over the heavier and the other will be a down-gliding surface
where the heavier water sinks underneath the lighter. The lighter water will gradually
extend completely over the heavier and will finally give a cyclonic vortex (in the top
layer) with a simple convergence point of the form a.

364

The Representation of Oceanic Movements nad Kinematics

(Q) / / (b)

(d)

Fig. 154. Singularities of the current field: (o) and (6) convergence and divergence point;

id) anticyclonic vortex at the boundary between two water masses.

The form d represents an anticyclonic vortex in the region between the two water
masses. Here the boundary surface sphts up into two divergence Hnes. In this case the
anticyclonic vortex causes a concentration of the hghter water in the central part of
the vortex. The dynamics of such cyclonic and anticyclonic vortexes will be discussed
later (see Chap. XIV, 4).

Neutral points (Fig. 155fl, b) occur when currents flowing in opposite direction meet
each other and separate again without showing stronger vertical motion. Two asymp-
totes to the stream lines then intersect at the neutral point situated in the centre.
Singular points of higher order are also possible. The current field is then very com-
plicated, see, for instance, Fig. 155a. In place of the second water mass there may be a
solid boundary at a coast line where the current divides into two parts. The neutral
point then lies on the shore line. In the presence of a wave motion the stream lines take
on a special pattern. During the propagation of a wave the individual water elements
usually describe elliptical orbital motions in a vertical plane perpendicular to the wave
front. The longer axis of the ellipse is horizontal and the smaller is vertical. For such
a periodic wave motion it is of course only sensible to plot a stream-hne pattern for a
particular phase of the wave motion. Fig. 155c shows that for a propagation of a wave
to the right the water masses will converge in front of the wave and will diverge in its
rear. At the sea surface this gives rise to a convergence line in front of the wave and a
divergence line behind it. Both the singular lines move with the wave at right angles to
the wave front. Thus in a wave motion two types of strips occur, so that strips with a
movement from left to right alternate with strips moving from right to left. If the wave
is propagated to the right the first type of strips will correspond to the wave crests

The Representation of Oceanic Movements and Kinematics

365

(c)

(d)

.^^^^^^^^s:^

(b)

(e) 'j

(f)

Fig. 155. Singularities in the current field: (a) neutral point, ib) one-sided neutral point,
(c), {d), (e) and (/) singularities in wave motions : (c) stream lines in a vertical cross-section,
(d) stream lines at the surface with a small translation parallel to the wave crests, (e) and (/)
the same with a somewhat stronger or a very strong translation oblique to the wave crests

(according to V. Bjerknes).

and the second to the wave troughs. If in addition to the wave motion there is also a
more or less strong translatory motion in the water mass, then the two current fields
will be superimposed on each other, and the resulting current field will consist of a
system of convergence and divergence lines moving parallel to each other with the
wave. Some fields of this type are illustrated in Fig. 155.

The singularities are closely connected with the velocity field. Where stream lines
intersect the velocity must be zero ; the points of convergence and divergence and the
neutral points must therefore be points of zero velocity (places of no motion). The
isolines of velocity must be closed around these points. When approaching singular
lines there will always be more and more curvature in the lines of equal velocity.
This curvature becomes stronger the more the stream lines converge towards the
singular line. For weaker convergences this curvature is usually hardly noticeable
in the observations.

Constructing stream lines usually offers little difficulty, especially if the position of
the singularities is fixed first. Usually some of the stream lines running out from the
singularities can be drawn in with some certainty and these fix the current field with
almost sufficient accuracy. Attention should also be paid, of course, to the velocity
field and to relationships with the dynamic phenomena expressed in the distribution
of other oceanographic factors (temperature, salinity, etc.). Sandstrom (1909) has
given a method for the accurate construction of stream lines. Auxiliary lines termed
isogons were drawn in first. An isogon is defined as a line along which the direction
of the current is constant, and for each direction there exists only one isogonal curve.
If the observed directions are expressed by numbers (usually 16 directions with the
numbers 2 to 32) then numbers can be entered on the chart in place of the arrows
indicating the direction of the current; the isolines of equal direction are then easily
constructed. These are covered with rather short dashes pointing in the direction of

366 The Representation of Oceanic Movements and Kinematics

azimuth of each isogen so that the chart is covered complete with short dashes. It is
then easy to draw in the curves tangential to these short dashes and these curves are
the stream lines. Werenskjold (1922) has pointed out that it is possible to draw in
a number of isogons rather quickly by simply using two charts of the eastern and
northern components of the current u and v. If a is the azimuth of the current then

tan a = - = k.
u

Each isogon is fixed by ^ = const. Two isogons can thus be drawn in immediately:
for A: = 0 and k = co; they correspond to lines y = 0 and m = 0. Their intersections
give the singular points through which all isogons must pass. Since the relation

V — ku = 0

is satisfied only at points where m = y = 0 for all values of k. Further isogons are
easily found; they can be limited to the eight isogons where ^ = 0, ij, ±1 and ±2;
corresponding to these are the azimuths 0°, 26|°, 45°, 63|°, 90° and so on. These
usually fix the current field with sufficient accuracy.
The stream lines are given by integrations of the differential equation

— = A: = ^
dx u

(see equation (X. 22) on p. 323). If v and u are given as analytical functions of the co-
ordinates X and y, then in many cases an accurate integration of the equation, and
therefore also a representation, of the current field is possible. Werenskjold has given
a large number of cases of this type and has discussed them in detail. Reference is
made to these. Of particular interest are those cases where complex singularities
occur; to draw these complicated patterns is usually rather tiresome, but mathematically
they are no more difficult than the simple ones. An example will illustrate this. If u
and V are given by

— u = x^ + (y + ay — r^,

V = x^ + (>' — ay + r^,

where r^ > a^, then the integration of the diff'erential equation above gives the stream
lines represented in Fig. 155a; the isogons u = 0 and v = 0 are circles which are shown
by dotted lines in the figure. Their points of intersection give the singular points, one
of which is a neutral point and the other is a convergence point ; they are connected
by a line of convergence. Such connections of the singularities are relatively frequent
in stream-line patterns of ocean currents.

(d) Examples of Current Charts

Current charts based on these principles have been prepared for many parts of the
ocean, usually for mean conditions since there are almost no synoptic data available.
They show only surface currents. Various types of presentation have been used. An
accurate representation based on strict hydrodynamic principles has been introduced
by Bjerknes. Analysis of the current fields and their resolution along the extended
lines of convergence and divergence, with more or numerous complex singularities,
has shown that the previous conception of large horizontal circulating systems in the

The Representation of Oceanic Movements and Kinematics 367

Fig. \55a. Example for a special current field according to Werenskjold (integral curves
of dyjdx =■■ v/u = k; circles are curves « = 0 (below) and t; = 0 (above)).

Fig. 156. Stream lines south of Africa for May (according to Merz).

368

The Representation of Oceanic Movements and Kinematics

currents of the surface layers is untenable, and that the deeper water layers are also
involved in the surface current systems. An example of this type of representation is
given in Fig. 156 which shows the surface currents to the south of Africa during May
according to Merz (1925). A line of convergence runs right across it separating the
steady broad west wind drift in the south from the Agulhas Current south of Africa.
Charts of this type do not indicate the velocity of the current, its prevalence or the
amount of data on which it is based; velocity is mostly indicated by thin dotted lines
(nautical miles per day). Because of gaps in the available data current charts such as
these, constructed according to strictly hydrodynamic principles, are naturally not
certain in all details, but the individual stream lines and singularities support each
other by means of their course and position and thus offer the clearest possible picture
of the water movement.

Another representation of essentially the same type was chosen by Helland-
Hansen and Nansen (1909, p. 9) in which the stream lines are represented by a
series of short arrows of more or less equal length (Fig. 157). Also here the velocity

LT — in

■^k^}

J^-x

Fig. 157. Mean currents at the sea surface of the European North Sea (according to Helland-

Hansen and Nansen).

The Representation of Oceanic Movements and Kinematics

369

- ,^ \ I .' E • \ /I v^ I t I t t «..•■••__ •

370 The Representation of Oceanic Movements and Kinematics

and the stability are omitted from these charts. This type of representation is usually
chosen for current charts which are based less on direct current measurements and more
on a qualitative assessment of the horizontal distribution of the temperature, salinity
and other factors which the system of currents at the sea surface must reflect.

A more comprehensive representation of the currents in an ocean has been used
by Meyer (1923) for an evaluation of Dutch observations on the currents in the At-
lantic during February. Figure 158 shows a part of this chart. Here also the stream lines
were broken up into a series of arrows of equal length but their thickness was used as a
measure of the current constancy (stabiHty), the feathering as a measure of the velocity
and the amount of data available was indicated inside the breaks in the shaft of the
arrow. The singularities in the current field are not shown so clearly by this type of
representation and are therefore indicated by special signs, particularly in the case of
the more important lines of convergence and divergence. These charts already permit
a deeper insight into the nature of the water movements at the sea surface of the ocean
under consideration and also allow an estimate of the reliability of the chart at any
particular area of the oceanic region. Similar but somewhat modified representations
have been chosen by Schott (1926, 1935) and Schumacher (1940, 1943).

In assessing the value of a chart and in its use, it is necessary to keep in mind the
relatively large uncertainties which still remain attached to them. The number of
observations on which the charts are based in the individual degree squares varies
considerably and is often so small in some of the squares that chance can be rather
important. These difficulties, however, may decrease with time since the number of
observations collected by hydrographic institutes increases from year to year and
mechanical evaluation of these data by computing machines is much faster than was
previously possible.

3. Special Cases of Current Fields Near Land and at the Boundaries of Water Masses
(Compensation Currents)

The boundaries of the sea, fixed either by coast lines or by the topography of the
sea bottom, exert a considerable influence on the pattern of the ocean currents and
especially on the form of the current field. For each steady current (potential flow)
the water in the immediate vicinity of a solid boundary surface (coast or sea bottom)
tends to approach the boundary as closely as possible. The effect of such disturbances
is thus shown rather far from the source of disturbance in the current field and also in
the distribution of the oceanographic factors (temperature, salinity, etc.). The most
simple cases, which occur also in nature again and again can be expressed mathe-
matically by the method given on p. 327 ; a few of these can be briefly treated here.

(1) Plane flow around a cylindrical obstacle (island) is given by a function
F = U{z + {a^lz)). Introducing in r = x + '>' polar co-ordinates, then

z — /-(cos (f) -}- i sin ^) = re^'l'

and the velocity potential O and the stream function ^ will thus be given by the
expressions

0 = t/ jr + -J cos 0 and "F = U Ir - ^-| sin cf,.
One stream line is the .v-axis for which sin <^ = 0, another one is the circumference

The Representation of Oceanic Movements and Kinematics

371

Fig. 159. Stream lines around a cylindrical obstacle (island).

of the obstacle where r — a^jr vanishes. The stream hnes of the potential flow are
given in Fig. 159.

(2) Choosing F = (fl/2)z2 then

0 = (a/2)(jc2 - j2) and W = axy.

The jc-axis and the >'-axis are stream lines (^ = 0) and one obtains in that way the flow
towards a straight and vertical coast at which the flow divides into two branches (see
Fig. 155 (b)).

(3) The function F = Az'^ leads to current cards for bays or around projecting land
masses where as a first approximation the boundaries can be taken as straight. Intro-
ducing again polar co-ordinates we obtain

0 = ylr" cos ncf) and 'F = Ar"^ sin n^.

Parts of the curves V = 0 can be taken as solid boundaries ; this leads to sin «^ = 0
or to the lines ^ = 0 and ^ = Trfn. Putting n = 77/a, then ^ = 0 and <j> — a,2a, . . .
can be taken one after the other as the sohd boundary. This gives the irrotational flow
(vorticity free) between or off two straight coasts which meet each other at an angle a.
Fig. 160 shows some cases which are of interest.

The configuration of coast lines and outer boundaries of ocean basins are con-
siderably more complex than in the simple cases which are susceptible to mathematical
analysis. The simple character of currents that carry water masses from a distance into
coastal areas will be disturbed and changed by the coast lines. An important role is

^=180° a--'^5° a--90°

^=270°

Fig. 160. Stream lines off a coast as shown on the picture (triangular shape).

372

The Representation of Oceanic Movements and Kinematics

played here by the compensation requirement which is a result of the continuity law.
Since water is almost completely incompressible it cannot accommodate a widening
or contracting of the stream lines by contraction or expansion and movements normal
to the flow direction or even counter currents are set up to a much greater extent than
in air movements in order to avoid the formation of empty space. The nature of these
counter movements can only be fully explored empirically by observations in nature
or by special suitable experiments. Experiments of this sort have been made extensively
by Krummel (191 1, p. 470) and have been used for a clarification of many phenomena
exhibited by the pattern of the ocean currents. The results of that shown in Fig. 161
are particularly instructive. The resemblance of the experimental current system to
that in the Central Atlantic can readily be seen ; this system consists of the two wind

Fig. 161. Experimentally produced current patterns (simulation of the current system in the
central part of the Atlantic Ocean) (according to Krummel).

drifts induced at the sea surface by air currents, and the corresponding circulations
to the north and the south as well as the (equatorial) counter current between them.
At the projecting peak on the left-hand side of the experimental tank representing
land (Cape San Roque) the current intensity was surprisingly large (corresponding to
the Guayana Current).

Standing vortices are formed at coastal bays, in which the flow always shows such
a sense of rotation that the current on the seaward side follows the main current while
that on the landward side is opposed to it. Hydrodynamically such a vortex can be
stationary, but it will always have the same water mass circulating within it and there
will be no water transfer from the main current to the vortex. In nature this is usually
not the case. Pulsations in the main current will always affect the intensity and the
extent of the stationary vortex and will thereby lead to a renewal of the water circulating
in it. Such replacement currents in bays and small gulfs are termed "neer currents"
and are always present at any reasonably irregular coast consisting of small bays and
projecting land. An example is shown in Fig. 162.

The compensation requirement need not always to be satisfied by horizontal trans-
ports, but vertical movements are also sometimes involved and give rise to very charac-
teristic oceanographic phenomena (upwelling).

Fig. 162. Sea surface currents in the northern part of Bosporus (according to Merz MoHer)
with stationary vortices in individual ba\s.

The Representation of Oceanic Movements and Kinematics

373

Fig. 162a. Two streams of water flowing together.

Conspicuous phenomena also occur where currents carrying two different water masses flow to-
gether and these deserve special attention. If two water masses of different type meet at a sharp land
projection or at a motionless water mass there will usually be an appreciable transverse velocity
jump at the boundary surface (Fig. 162a). It cannot be expected that separating surfaces of this
type will keep for any length of time their simple form, since the state under consideration is highly
unstable. Every boundary surface of this sort has a tendency to develop waves and all chance
irregularities will thereby grow rapidly and the discontinuity surface will finally dissolve into a
number of irregular vortices. These processes are particularly characteristic for the transition from
waves to vortices and have been described in detail by Bjerknes (1933) and Prandtl (1942).
A boundary surface at which a temporary disturbance of the current field has given rise to a slight
bulge is shown in Fig. 1626. This wave-form disturbance will move along the boundary surface with
the average of the speeds of the two currents; relative to this wave one of the water masses
will move to the right and the other to the left, and with reference to this kind of co-ordinate
system the ridges and troughs of the waves will remain in the same place. According to the
Bernoulli theory the disturbance in the course of the stream lines will be accompanied by a
corresponding transverse pressure disturbance. For a steady state of motion the transverse
pressure rise llpidpjds) must be balanced by the centrifugal acceleration c^/r (c denotes the hori-
zontal velocity, r the radius of curvature of the stream lines, s the direction of the normal to the stream
lines). It becomes obvious that there will be a pressure surplus (+) in the ridges of the waves and a
i educed pressure (— ) in the troughs of the wave. This implies that the wave disturbance cannot be
stationary but that the water begins to move from the surplus pressure areas to the adjacent areas of
reduced pressure; that is, as the wave disturbance becomes stronger it will form current fields similar
to those in Fig. 162c, in which the boundary surface will finally be rolled up into vortices, lying one
behind the other and all rotating in the same sense. The same phenomenon occurs here in the hori-
zontal plane between two water masses with different velocities as in the case of unstable waves at the
boundary surface between water masses flowing one above the other (see Vol. n. Chap. XVI, p. 517
Internal waves). Examples of cases such as this are the vortex formations at the boundary between the
East Greenland Current and the Atlantic Current in the Irminger Sea, or the vortex-formations at the
boundary between the Gulf Stream and the Labrador Current south of the Newfoundland Banks (see
p. 471).

— >■
' — >■
— ^
— >■
— >■
■< — — >J

■* —

Fig. 1626. Disturbances in the pressure field due to wave-like deformations of a boundary

surface between two currents.

374

The Representation of Oceanic Movements and Kinematics

rrrTTTTTrr.
Fig. 162c. Formation of eddies behind a sharp edge and their growth.

4. Divergence of the Current Field and the Continuity Equation

The current field for a horizontal movement can give information about the place
where vertical water movements must occur within the field. Since, on the one hand,
in an incompressible medium, divergent and convergent stream lines must be asso-
ciated with vertical displacements and on the other hand for parallel stream lines,
velocity changes will lead to water accumulations (piling up of water; "Wasser-
stauungen") which will also cause vertical movements. Quantitative relationships can
be derived from the following considerations.

If A A' and BB' in Fig, 163 denote two adjacent stream lines, ds and ds' are elements
of these, c and c' are two lines of equal velocity in the current field and 8n as well
as 8n' are the parts of these lines between the stream lines, then it is possible to calcu-
late the amount of water flowing through the small area ABA'B' — ds 8n in unit
time. This outflow per unit area is termed the divergence of the current field and is
indicated by div c. It is a measure of the divergence and convergence of the stream
lines and also of the velocity. One therefore obtains

div c =

dsSn

[c'B„' - an] = I + I, f = ^ I (^ ««). (XII.I)

If the velocity along the stream lines is constant (c — const.) and the small angle be-
tween the tangents to the two adjacent stream lines is denoted by 5a then the curve
divergence is given by

c dSn 8a

div c = ^ -^ = c Y-'
on OS on

(XII.2)

The Representation of Oceanic Movements and Kinematics

375

Fig. 163. Divergence of the current field.

The divergence is positive if the stream lines move apart and negative if they contract.
If the stream lines are parallel {hi = const.) then

div c —

8c
Ts

(XII.3)

The divergence here is a consequence of the change in velocity in the direction of the
stream lines; a decrease indicates pihng up ("Stauung") and an increase indicates a
suction of the water masses.

For a given current field the divergence field can be calculated numerically or gra-
phically and can be represented on charts; special methods for this have been given
by Bjerknes and co-workers (1912, 1913).

The general continuity equation (X. 22) can be written in the form

dp
dt

+ p div c = 0

for an incompressible water mass this gives

div c = 0.

(XII.4)

(XII.5)

If allowances are made for changes in density due to changes in temperature and
salinity, then equation (X. 21) applies and for stationary conditions one obtains:

dpu 8pv 8pw

8x 8y 8z

div c = 0.

(XII.6)

The total horizontal water transport ("current amount") in a water column from the
surface (z = 0) to the bottom of the sea (z = h) is then

A/ = pc dz
Jo

and its components along the x- and j^-axes are given by

(XII.7)

pu dz and My =

pv dz

(XII.8)

376 The Representation of Oceanic Movements and Kinematics

Multiplying equation (XII. 6) by dz and integrating from the surface to the bottom it
follows that

At the sea bottom w^ equals 0 and further if the vertical elevation of the sea surface
above the equilibrium level (positive upwards) is denoted by i then Wq = —{dlidt)
and from (XII. 9) follows

8t 1 ,

-f= divAf (XII.IO)

dt po

The divergence of the current amount is thus always associated with vertical displace-
ments of the sea surface and these can be readily calculated from (XII. 10) if the current
amount is known. For a stationary state of the sea surface (C — const.) it follows
necessarily

divA/ = 0, (XII. 11)

that is, at stationary sea surfaces the total current amount must be divergence free.
This need not be the case in every layer but in the entire water column an excess in-
flow in some of the individual layers must be balanced by a deficit in the other layers,
if no effect on the sea-level should appear.

Under stationary conditions in the sea there must be in any volume element a con-
stant amount of all the dissolved substances in the water besides the constancy in
density (see Defant, 1941^/). If the salinity for example is denoted by s and exchange
processes are for the moment disregarded, tliis requires

ds ds 8s 8s ^s ^ ^ , ,^.

^. = ^ + " ^ + ^ ^ + '^' TT = 0- (XII.12

dt dt 8x 8y 8z

Multiplying this equation by p and then adding the continuity equation (X. 31)
multiplied by s, it follows that

8 OS 8ups 8vps dw'ps

For stationary conditions the first term on the left-hand side is zero and the condition
of a constant salinity will be given by the remaining equation integrated over the total
volume under consideration. Introducing a space vector S with horizontal components
Sx and Sy which is given by the equation

•/I
S= pscdz (XII. 14)

J 0

allows the equation (XII. 13) for stationary conditions to be rewritten in the form

div5 = 0 (XII. 14a)

S can be termed the salinity amount and the equation states that under stationary con-
ditions the vector indicating the amount of salt flow must also be divergence-free.

The constancy of the water mass in a given space and the constancy of the characteristic water
properties existing under stationary conditions has often been used in the derivation of the current

The Representation of Oceanic Movements and Kinematics yjl

amount in the considered space. For example, the silicate content is q at three oceanographic stations
a, b and c, where the vertical salinity distribution is s. For a prism taken by these stations down to a
definite level, there will be current amounts M^, Mj, M3 passing through each side in unit time and a
current flow M^ through the bottom surface. If it is then assimied that no water enters or leaves through
the upper sea surface (zero precipitation and evaporation) then the constancy of the water volume
requires that

Ml + M2 + M3 + M„ = 0.

If further the corresponding mean amounts of salt and silicate passing through the three surfaces of
the prism are indicated by s^, s^, s^ and q^, q<i, q^, respectively, and the amounts of salt and silicate in
the prism are taken as constant, then

s^Mi + S2M2 + .ygMg + s^M^, = 0
and

^iMi + q^M^ + q^M^ + qJA^ = 0.

If the current amount or the current at one of the lateral surfaces of the prism are known the three
equations are sufficient for a calculation of the other three unknown currents.

Okada (1934) has used these methods to study the oceanographic conditions in the Sagami Bay;
and they have been used in a more extended form by Hidaka (1940a, b) to reduce the relative velocity
distribution calculated from the oceanographic structure at different stations to the absolute values.
Unfortunately, however, these methods cannot be used in most cases just for numerical reasons, since
the coefficients of the equations diffisr numerically by so little that the determination of the un-
knowns becomes illusory. In the second and third of the above equations the mean salinity and
silicate values at the three surfaces of the prism differ very little, so that the equations are only in-
significantly different from the first. Small errors in the determination of the values of 5 and <? and other
random effects such as inaccuracies in the positions of the stations thus play such an important part
in the solution that no reliance can be put on it.

In using the continuity equation for the determination of the current amount it
should be borne in mind that the distribution of the characteristic water properties is
largely controlled by exchange processes, so that these cannot be neglected since the
magnitude of these effects is the same as that of the simple transport terms. To be
strictly correct the equation (XII. 12) should also take into account the effects of
mixing processes. This leads then to an equation which has already been used in
Pt. I (see p. 120) in the explanation of the phenomena occurring during the spreading
of a water mass into surrounding waters. For stationary conditions it takes the form

dpus dpvs 8pws d / 8s\ 8 / 8s\ 8

8pvs 8p\vs 8 1 8s\ 8 1 8s\ 8 [ 8s\

Integrating this equation from the sea surface down to the sea bottom the last term
on the right-hand side gives

8s \ I 8s

'k-

The first term of this expression is zero since A^ vanishes at the sea bottom. The
second represents the difference between evaporation and precipitation per unit area
at the surface of a water prism.

Neglecting the effect of the vertical component of velocity on the left-hand side of equation (XII. 15)
on account of its smallness and retaining on the right-hand side only the term for the vertical exchange,
then for p Y^ 1 and A^ = const, an approximately correct equation is obtained

ds ds d'^s

378 The Representation of Oceanic Movements and Kinematics

which has been used by Okada, 1935; Thorade, 1935, in a graphical procedure for the investigation
of currents. Integrating it from z = 0 to a depth z = h and replacing in a first approximation the in-
tegral on the left-hand side by the mean value of the individual quantities (indicated by a bar over the
symbol) then the following expression results

-•--[(a -(!).]■

ds -ds A.

Taking the x-axis in the direction tangential to an isoline so that dsjdx = 0 then, since dsjdy is in-
versely proportional to the distance D between two isohnes, the current component v perpendicular
to the isoline will be given by

Z=^[(ll-(i)o]-

The expression in brackets on the right-hand side can be determined from observations and the velo-
city component can therefore be obtained. Lines of equal silicate content will in the same way give a
second velocity component across these lines and finally afford an estimate of the total mean velocity,
provided A is known by other means. Accurate determination of the isolines is, however, an essential
presumption in the use of this method.

For a homogeneous sea with a homogeneous current structure the relationship
(XII. 10) {u and v independent of r) takes the simple form

di (du dv\

It can also be readily derived from the continuity equation. It can be used to judge
the accuracy with which the vertical mass transport can be deduced from the distribu-
tion of the current flow vector. Thereby it shows immediately that for its evaluation
the deeper the sea the more accurately the horizontal distribution of u and v must be
known. The use of this relationship is thus hmited to shallow shelf seas. Here, par-
ticularly in representations of tidal currents, it allows the corresponding vertical
tide to be deduced (Defant, 1925). If c is the velocity of the tidal current

C = Cq cos {at + e),
^ = ^0 sin (at + e),
then using (XII. 1) the relation (XII. 16) can be given the form

8i h 8

Insertion of values for c and ^ gives the equation

which is independent of the time. Now the following cases may occur (see Fig. 1 63)
(1) Parallel stream Unes

8n = const, and ^0 = ■ ^-.

a OS

Assuming Co = 100 cm, for the distance 8s between two stations 50 km and for
h = 50 m, then one obtains for the semidiurnal tide (or = Itt 112-3 h) the necessary

The Representation of Oceanic Movements and Kinematics

379

Scq = —14 cm/sec = —0-25 nautical miles per hour. This horizontal change in the
maximum velocity component is well within the accuracy of measurement.
(2) Divergent stream lines for a constant velocity (cq = const.) :

hCff 8a
a 8n'

For identical a, l,^ and h and taking again the distance between two stream lines,
8n = 50 km and Cq as 50 cm/sec one obtains — Sa = 0-284 angle units or about 16
degrees of arc. This divergence is usually easily readable from charts of tidal currents.
The method thus gives results of sufficient accuracy provided the ocean depth is not
too great ; for example, it has been apphed successfully to the evaluation of the tidal
conditions of the North Sea (see Vol. II).

Where the structure of the sea has two or more layers a relationship of the form
of (XII. 16) can be derived for each boundary surface between two successive super-
imposed layers. These relations fix the time changes in the inclination and position
of the boundary surfaces as a function of the divergence of the currents in the indi-
vidual layers.

5. The Knudsen Relations

The relations for the current amount (XII. 1 1) and for the sahnity amount (XII. 14)
allow an insight into the current conditions in more or less exactly limited oceanic
regions such as sea straits and river mouths and others. Knudsen (1900) derived some
simple laws of this type which are based fundamentally on these relations and just
because of their simplicity and clearness lead directly to valuable conclusions about
the general current conditions in such areas.

In straits, river mouths and also in the open ocean lighter (low saline) water often
spreads out over heavier (more saline) ; in such cases the currents in the two layers
are mostly of opposite direction. In Fig. 1 64 A and B are two vertical cross-sections

Fig. 164. Water and salt transport through sea straits.

through such an oceanic region (for instance a strait). If the mean salinities are s
and s' at A and B in the upper current and z and z' in the lower current and the
current amounts are / and /' in the upper current and u and u' in the lower, then, under
steady conditions the constancy of water and salt transport in each current will give
the equations

I = u

u ; IS = U2

I s = u z

380 The Representation of Oceanic Movements and Kinematics

If the mean salinities are known these relationships give the Knudsen relations in the
form

z' z — s s s' z — s _

/' = / - -, ,; u = i J u = i- -_-, ;. (XII.19)

z z — s z z z — s

If the upper current is known at one point and the distribution of salinity is known
at two, these relations allow an evaluation of the mean water transports at different
cross-sections in the strait; if no current amount is known they give at least the inflow
and outflow conditions which in itself is valuable.

If section A is taken so far inland within the river mouth that only fresh water is
present {s = 0, as well as z and u = 0), then it is the mean water amount carried by
the river seawards per second and from (XII. 19)

/' = / -A — ^ and u' = / -j^ — ;. (XII.20)

A longitudinal section given by F. L. Ekman for the Gotaelf showed s = 18,2' = 22%o
so that /' = 5-5 7 and u = 4-5 / that is /' \u' = \\ : 9. The thickness of the upper cur-
rent was 3 m, that of the lower current was 9 m and thus in one second there was a
flow of 1 1 volume units per unit area seawards in the upper current compared with a
flow of 3 units upstream in the lower current.

Another example given by Knudsen refers to the Baltic. Cross-section B\ cross-
section through the outlets of the Baltic (the Oresund and a section from Gedser to
Darsserort) ; cross-section A : the entire surface of the Baltic and sections through
all the river mouths. Here / in (XII. 21) was the entire amount of water entering the
Baltic per sec due to precipitation, evaporation and run off by rivers. From the salinity
distribution is obtained s' = 8-7 and z' = 17-4%o from which it follows that /' = 2/
and u' = i. Thus the upper current carries twice as much water out through all the
outlets of the Baltic as is carried into the Baltic by the lower current and the amount
of water flowing in with the lower current is equal to the actual inflow from other
sources (precipitation, evaporation and river water). Since /' = / + u' only half the
outflow is derived from fresh-water gain, the other half is balanced by the inflow in
the lower current from the sea.

In a third example Knudsen placed the cross-section A through the Oresund and the Kadet-channel
and cross-section B through the Kattegat from Fomas to the Skalle Riff. K\A,s= 8-7 and z = 17-4%o;
however, at B, s' = 20 and z' = 33%o. With these values the relationships XII, 20 give /' = 1-27/,
u' = 0-77 / so that /' : u' = 1-65.

The amount of water flowing out through the Kattegat section is about 5/4 times greater than that
flowing in from the actual Baltic Sea through the Oresund and the Kadet-channel, that is, it is about
2-5 times larger than the total gain of the Baltic in fresh water. It is also found that the amount of
salt water flowing in into the Kattegat from the south is about 1-5 times larger than the amount of
salt water flowing in into the Baltic. This amount of water is the same as half the entire inflow into
the Baltic and indeed penetrates into the western part of the Baltic but mixes with the upper current
and is carried out again.

A further example is given by the oceanographic conditions in the Bosporus. At a
cross-section at the south-west end salinity measurements (September-October 1917
and May 1918; Moller, 1928) gave . = 37-65 and s = 17-47%o with the boundary
surface at a depth of 23 m ; however, a cross-section at the north-east end gave z' =
35-79 and .y' = 17-23%,,, the boundary surface depth being 44 m. From (XII. 19)

The Representation of Oceanic Movements and Kinematics 381

these values gave the relationships (calculated) /' = 1-03 /, u = 0-465 /, u = 0-449 /
and u = 1-07 u. Current measurements gave the relations (observed) /' = 1-06 /,
u = 0-55 /, u = 0-449 / and u' = 1-22 u.

Considering that the observational values were obtained from only a few individual
measurements and that meteorological factors have an appreciable influence on water
transport through the Bosporus the agreement is very satisfactory.

A generalization of the Knudsen relations has been given by Witting (1906) in
his study of the Gulf of Bothnia ; here an attempt was made to consider in the calcula-
tion changes in sahnity during the observational period, although one is mostly forced
by the observations to be satisfied with the simpler equations (XII. 19). The investiga-
tions of Gehrke (1907) on the current conditions west of Ireland and the British Isles,
where the Atlantic current flows north-east were also based on these relationships;
they are an example of how similar ideas can be applied to corresponding problems
concerning the oceanic circulation, even then, when no upper and lower current flowing
in opposite directions are present.

All these investigations are based entirely on the continuity equation for the water
and salt contents and of course yield information only on mean conditions ; they do
not give any information about the internal structure of the c urrents or on the causa-
tive connections between them.

Chapter XIII

General Theory of Ocean Currents
in a Homogeneous Sea

1. Introduction

A THEORY covering all the phenomena of ocean currents and taking into account ail
the effects of the internal and external forces must essentially be rather complex and
would not allow an immediate insight. The theory must thus, as in other fields of
natural sciences, take another path as soon as it can be based on well-founded geo-
physical principles, and must use simplifying assumptions taking only the effect of
one single current-generating factor into account at a time. All these individual
current constituents can then be combined to give some picture of all the factors in-
volved in the generation and maintenance of the ocean currents. This, of course, is
the aim of any theory; because of the complexity of the phenomena involved, little
could be deduced from the dynamics of the ocean currents developed in their most
general form that would assist in the elucidation of the nature of the oceanic circula-
tion. The history of the theory of ocean currents is long and goes back a long way and
would require considerable space; a more or less detailed account of the older parts
has been given by KrOmmel (1911, pp. 442-449). The first simplification is the ehmi-
nation of the internal forces; this is identical with the assumption o^ a. homogeneous sea.
In this case only external forces would be able to produce water movements. Ekman
(see especially 1927) was the first to develop the problems of the dynamics of the ocean
currents of a homogeneous sea in a classically elegant form and went far towards
successful solutions for these. There are two immediately apparent problems :

In a homogeneous sea, movements of the water may arise besides from the effect of
the wind on the sea surface also from the pressure of a sea surface slope. This gives
rise to a horizontal pressure gradient which is transmitted through the entire water
mass down to the bottom. The first main problem is then the calculation of the velo-
city components at each level for a given wind force and a given gradient of the sea
surface. The hydrodynamic equations of motion provide the basis for this and can be
solved, as has been shown by Ekman, if the frictional coefficient is given. The current
system produced by the action of these external forces at all points along a vertical
was termed by Ekman the ^'elementary current.

The constituents of the elementary current can be derived without taking the con-
tinuity equation into account. Due to differences from place to place in the wind distri-
bution or the sea surface slope or due to local differences in the depth of the sea the
continuity requirement cannot be satisfied by horizontal movements alone. The di-
vergence of the currents caused in this way gives rise to changes in the sea-level which
in turn affect again the elementary current (feed-back). The second main problem

382

General Theory of Ocean Currents in a Homogeneous Sea 383

consists only in following these changes in the elementary current or in determining
under stationary conditions the elementary current that satisfies the continuity equa-
tion, and then in evaluating the associated time-independent sea-surface slope for all
points of the oceanic region under consideration. Only then can the problem be
considered as completely solved. This second problem is the more difficult one since
the boundary conditions at coasthnes must also be satisfied. It does, hov/ever, help
to produce the total picture of the currents for a certain preassumed ocean basin.

The starting equations for the development of the dynamics of the ocean currents
are the hydrodynamic equations of motion in their most general form (see equation
X.16). The fact that its individual terms are of quite different significance led Jeffreys
(1922) to put forward a terminology for air currents which could also with advantage
be applied to ocean currents. According to whether the horizontal pressure gradient
is balanced principally by the acceleration or by the Coriolis force or by friction, it is
possible to distinguish between (equations for the .v-axis only, those for the j-axis
being analogous):

du 1 dp

Euler current : -y — 7r\

at p ex

geostrophic current : 0 = ^ — h 2a> sm ^y ;

... ^ \ dp d / 8u
antitnptic current :0= ~ — \- — [a ^r

p ox cz \ S.v

The Euler current will appear for rapid changes in the sea level (storm surges, etc.) ;
this is also the relationship on which is based the simple theory of waves, where the
water displacements in general have the character of a Euler current. The geostrophic
current corresponds to another current constituent of the "elementary" current,
namely to the gradient current (deep current), while, during the formation of the wind
drift and the bottom current, besides the Coriolis force to a considerable extent fric-
tion is also involved. An antitriptic current can be expected in local circulations of
small extent, for example, in equalization currents in sea straits where the narrow width
prevents an effect of the Coriolis force.

2. Steady Currents in a Homogeneous Sea Without Friction

(a) General Equations

For a horizontal frictionless water movement, the equations of motion (X.16) for a
homogeneous sea (p = const.) (Coriolis parameter/ = 2aj sin ^) will take the form:

du ^ \ dp dv \ dp ■ -■• ■ -

-7: =>--/; TT. = -/«--/• (XIII.l)

dt p dx dt p dy ^

In a homogeneous sea the pressure p at a depth z (counted as positive downwards
from the undisturbed sea level r = 0) is given by

. p = gp(z + 0, , (XIII.2)

384

General Theory of Ocean Currents in a Homogeneous Sea

where C is the elevation of the sea surface above the undisturbed level (counted
positive upwards). Equations (XIII. 1 and 2) then give

and the condition for non-accelerated (stationary) current is then

(XIII.3)

(XIII.4)

or if the total velocity V = ^y{u^ + y^) and d^fdn is the total pressure gradient {n
normal to the lines of equal water level)

fdn

(XIII.5)

For a steady current pressure force and Coriolis force will be in equilibrium. Fig. 165
shows diagrams of the forces acting on such currents for both the Northern and the
Southern Hemisphere. The currents follow the lines of equal water level which are at
the same time isobars on the level surfaces ("Niveau-Flachen") and it follows the
proposition: In the Northern Hemisphere when facing downstream for a steady friction-
less water movement the higher water level will lie on the right-hand side of the current
direction and the lower water level will be on the left-hand side; the slope of the sea surface
is a measure of the current intensity. Such a current is termed a geostrophic current.

Lower water level

o
o

Gradie

^ Current

Higher water level

Lower water level

Current

Higher water level

Fig. 165. Schematic distribution of the forces for a stationary current in a homogeneous
ocean without friction (left side: Northern Hemisphere; right side: Southern Hemisphere).

Equation (XIII.3) permits integration if the topography of the sea-level is constant
in time or unchanged by the current. Multiplying the first equation by u and the second
by V, and adding, gives the relation

¥V^= -gdl
If a small water particle moves along a level surface from a point where the sea level

General Theory of Ocean Currents in a Homogeneous Sea

385

is ^0 above the equilibrium level, to another point where this deviation is ^i, it will
acquire a final velocity V^ given by the relation

V,^ = 2g(Co - Ci) (XIII.6)

if it was at rest at the starting point {Vq = 0). Corresponding values of Fj and ^o — ^i
are given in Table 115.

Table 115

$Q - ^1 (mm) . . 1

5 10

100

150

Po ~ Pi (centibars) 001
Vi (cm/sec) ; 14

002
20

005
31

010

0-50

100 1-50
139 312

If a water element glides downwards without friction along an oblique pressure
surface through a short vertical distance, it will immediately acquire a very large
velocity. If the water masses were not forced by the Coriolis action to move along the
lines of equal water level under stationary conditions, even a very small slope would be
able to cause enormously intense ocean currents. Equation (XIII. 5) shows that the
forces producing the movement {gradient force) do not, in the stationary case, determine
the acceleration of the water movement, but solely, due to the Coriolis force, its velocity.

(b) The Effect of Changing Depth and the Spherical Shape of the Earth

Equations (XIII.4 and 5) show that the entire water column down to the sea bottom
will have the same velocity; it will move hke a solid body with a velocity V in the
appropriate direction. This current can only satisfy the continuity equation if the sea
bottom is plane. Under stationary conditions {dijdt = 0) according to equation
(XII. 16) the continuity equation takes the form

dx 8y

= 0.

(XIII. 7)

It will be satisfied by the values of u and v given by (XIII.4). At constant depth there
will thus be no limitation to a geostrophic current. If there are boundaries to the sea
in the form of vertical coasts then the boundary condition will require a constant C
along them ; the current will then flow only along the coast and there will be no flow
perpendicular to the coast.

If the ocean depth is variable, conditions will be more comphcated. In Fig. 166
is shown the case where a given uniform slope of the sea surface (Northern Hemisphere)

Constant

Decreasing

Increasing

water rtootti

water deptti

water depth

Fig. 166. Deviation of ocean currents for a variable bottom depth.

386 General Theory of Ocean Currents in a Homogeneous Sea

from the surface of the figure backwards gives rise to a uniform current from left to
right; at first there will be an equilibrium in it between the gradient and Coriolis forces.
If the depth of the sea increases in the current direction (bottom slopes downward)
then for a constant flow amount, since the current cross-section becomes larger, there
must be a decrease in velocity. The equilibrium between the two forces will be disturbed,
the lower velocity attained will correspond to a smaller Coriolis force and the current
will be deflected contra solem. However, if the depth decreases (i.e. the bottom rises)
the velocity must increase; this will give an increase in the Coriolis force and a deflec-
tion of the current cum sole. The equihbrium state of equation (XIII.4) will continue
for each stream line only when the current follows the depth lines of the bottom.
If the depth is variable, (XII. 16) will be replaced by the continuity equation

di (dhu 8hv\

Under stationary conditions the equations of motion (XIII.4) will then give the con-
dition

8h dC 8h dl

This relation states that if the depth varies then steady frictionless currents are only
possible if the topography of the sea surface on a relative scale accords with that of the
sea bottom. The currents must thus run parallel to the bathymetric curves; the strength
of the current is, however, free and depends only on the absolute gradient of the
^-values. If there are coastal limits, the boundary condition requires that the depth
should be constant along the outer boundary (the coast).

Since the continuity equation for currents in an ocean partly or completely covering
the spherical Earth has a diff'erent form (equation (X.27), the conditions for steady
currents will also be different. The equations of motion for the meridional and zonal
velocity components will now be {R = Earth radius, & = 90° — ^ = zenith distance):

g ^l g^l

U = — 75-^ 5 ^Y ^"^ ^' = fD aQ- (XIII. 10)

fR sm § 8A fR dd

For a variable depth // and taking into account that h is always small compared with
7?, the continuity equation will have the form

dl 1 Idh sin du dhv\

The condition for a frictionless steady current is then under these conditions

8h 8i 8h 8t 8t

The first two terms are identical with the condition for planar co-ordinates (equation
XIII.9); they thus include only the efl'ects of variable depth. The third term
h tan d{8t,j8X) takes into account the eff'ect of the spherical shape of the Earth; it is
largest in the equatorial regions (§ close to 90°) and vanishes at the poles {d = 0°).
Some special cases can be selected to illustrate the two efl"ects.

(1) If the depth of the sea is constant, the conditional equation is satisfied only if
8l,j8X = 0, i.e., only if zo«a/ currents are possible (along latitude circles).

General Theory of Ocean Currents in a Homogeneous Sea 387

(2) The depth shall be a function of the latitude only. Then dhldX = 0 and the topo-
graphy of the sea bottom will be symmetrical about the poles. In that case, according
to (XIII. 12), there must be either dijdX = 0 or chjcd + /? tangi^ = 0. The first condi-
tion leads again to zonal currents ; the second gives on integration h = H cos d where
H is the depth of the sea at the poles {d = 0°).

In these cases both d^jdd and dl,jcX are free, that is, I, is also free. For a meridional
depth distribution of this type (decreasing gradually from a depth H at the poles to
a depth zero at the equator) steady currents would be possible in any direction also in
an ocean on the spherical Earth; conditions here are then the same as in a sea of
constant depth with planar co-ordinates. For this depth distribution both effects
balance exactly. It can therefore be deduced that in higher latitudes small changes in
depth will be able to compensate the effect of the curvature of the Earth, this effect
will therefore be small there. On the other hand, in lower latitudes larger changes in
depth will be required to balance this effect and therefore almost only zonal currents
will be possible. The critical vertical gradient in meridional direction which will be
able to balance the effect of the spherical shape of the Earth is given by {hIR) tan (^.
Table 116a gives these critical values for different latitudes and for depths of 3000 and
5000 m.

Table 116a

Polar distance .
Latitude

20°
70° !

30° 40° 1 50°
60° 50° 40°

60°
30°

70°
20°

80°
10°

Critical bottom
gradient for
h = 3000 m
h = 5000 m

1:5810
1 : 3500 :

1:3670 1:2540 1:1780
1:2190 1:1520 1:1070

1:1220

1:735

1:773
1:464

13:73
1:224

The discussion of the above equation (Defant, 1929fl, p. 61) leads to an estimate of
the two effects on steady currents. Following Ekman (1923), these can be summarized
as follows : Up to 3-4° latitude — and when the changes in depth are small, even farther
away from the equator — the effect of the bottom relief is rather unimportant for the
tendency of the current to flow in zonal direction. Between 10° and 20° of latitude
the two effects are equal and in higher latitudes (> 40°) the effect of the bottom topo-
graphy gains in importance and the currents tend to follow definitely the isobaths of
the sea bottom. The observed fact that in reahty ocean currents do preferably follow
a zonal direction in lower latitudes and their direction in higher latitudes is presumably
more affected by the bottom topography, appears to be reasonably well explained by
the thf oretical results presented above,

3. Eddy Viscosity (Turbulent Friction) in Ocean Currents

(a) Mixing Length and Eddy Viscosity (Turbulent Frictional) Coefficient

The movement of the water masses in ocean currents is mostly disordered and tur-
bulent and part of the strong variations in speed and direction of the flow which are
observed in quick-response recordings (see p. 347) can be attributed reasonably to this
internal turbulence. More or less large elements of water (water quanta) are continu-
ously being carried by these internal turbulent motions into the layers above, below or

388 General Theory of Ocean Currents in a Homogeneous Sea

to the side and there is thus an equalization of the momentum (current impulse) in the
direction of the strongest velocity gradient. There is also an associated equalization
of all the characteristic substances and of the water properties. This equahzation pro-
cess has already been discussed in detail in Pt. I, Chapter II (see p. 105). For the
property-pair momentum-velocity under conditions of immediate and complete
equalization of the flow momentum a general expression for the apparent shearing
stress of a turbulent flow has been derived having the form

(XIII. 13)

where U is the mean velocity along the x-axis, :: is perpendicular to it, t] is the exchange
coefficient for momentum (eddy coefficient or turbulent frictional coefficient).

In Chapter II (see p. 329) another expression was derived for the apparent shearing
stress occurring in turbulent flow from the analysis of the current variations in it.
This was given as

T=-pi7^"'. (XIII. 14)

The variations in velocity u' and v' are of course connected with the distribution of the
mean velocity which varies across the stream lines. To give a practical form to equation
(XIII. 14) Prandtl (see especially 1942) introduced the mixing length I, defined as the
length which can be regarded as the diameter of the water quanta moving with the
turbulent flow or as that distance that such a quantum travels before losing its identity
due to mixing with the surroundings. A water element with a mean velocity u(z) at a
point z (see Fig. 167) will have a mean velocity u(: + /) = m(z) + l{8uldz) at a distance

777777777777777777777777777777777777777777.

Fig. 167.

/ across the current. If a water element is moved from one layer to another then the
magnitude of u is given by

u' = u(z + /) - i7(r) = l{du\dz).

The variations in velocity v arise from the movements of the water elements entering
the place under consideration from different sides, moving one behind the other and
approaching or receding from each other with a velocity diff'erence of ll{du\dz) and
thus give rise to transverse movements. Thus r' will also have the order of magnitude
l(dul8z). Between u' and v' there must, however, be a negative correlation. The water

General Theory of Ocean Currents in a Homogeneous Sea 389

elements entering from below will have too small a velocity, those entering from above
will have correspondingly too large a velocity as compared with the velocity at the
point under consideration ; positive v' will thus occur together with negative u' and vice
versa. The product ii'v' is then always negative. The apparent shearing stress is thus
always positive and of the order of magnitude p{I(8ilj8zy}. The proportionality factor
is here arbitrarily taken as 1 ; this means only a slight change in the meaning of /. To
express in this relation that positive cii/cz will accompany a positive shearing stress
and negative ciijdz corresponds to a negative shearing stress, the eddy stress must
be re-written in the form

cii

= pP

— . cxin.15)

These turbulent shearing stresses change proportional to the square of the velocity and
this has been shown experimentally in investigations in hydraulics. The mixing length /
is not a constant here, but depends on the conditions in the current and will vary from
place to place. At a solid boundary it is zero and increases with distance from the
boundary.

Comparison of the two equations (Xin.13 and 15) leads to

cil

= pP

(XIII. 16)

The eddy viscosity coefficient depends not only on the mixing length / but also on the
velocity and density and is thus less susceptible to clarity than the concept of mixing
length. However, oceanic turbulence problems can only be handled numerically using
the quantity t], the eddy viscosity coefficient, especially for a freely developed turbu-
lence remote from solid boundaries (coasts and sea bottom). In the layers near the
bottom, however, there are considerable advantages in the introduction of the mean
mixing length as a characteristic number giving the degree of the turbulence as a
function of the distance from the bottom and of its roughness.
From relation (XIII. 15) it can be seen that the quantity

V P

cu
l—^ (XIII. 17)

has the dimension of a velocity. It is termed the friction velocity (shearing stress velo-
city) w,, so that T = puj which as mentioned above gives the flow resistance as a
quadratic function of the velocity.

The behaviour of a turbulent flow above a rough surface can be judged upon using
equation (XIII. 17), making an assumption about the mixing length / (Prandtl,
1942, p. 108). Since /increases with the distance from the underlying surface (z = 0),
it can be put equal to kz and if w, is constant, (XIII. 1 7) gives the solution

u = u, (-Inr + c). (XIII.18)

As has been shown in numerous investigations, the observed profiles are rather well
approximated by such logarithmic velocity profiles; for the number k the universal
value 0-40 was obtained. If ordinary decadic logarithms are used instead of natural
ones, equation (XIII.18) becomes

M = 5-75M, log -. (XIII. 19)

390 General Theory of Ocean Currents in a Homogeneous Sea

This represents a rather simple connection between the friction velocity and the
actual velocity distribution above the bottom. The integration constant Cq can be
related to a roughness length or parameter k. It has been found that for small bottom
irregularities such as occur on a flat bottom, sand or snow surfaces or surfaces with
not too large plants Cq can be given the value Cq = (A:/7'35), where k is the average
roughness parameter corresponding to the irregularities. If the bottom irregularities
are very large, it is difficult to determine the position of the point where z = 0 for
which the mixing length should vanish. It is then best to shift the zero point upwards
by a distance Zq and to use z -\- z^m place of r in equation (XIII. 19). This will then
mean that in the space within the major irregularities the mean height of which is Zq
the turbulent mixing length falls very rapidly to zero.

The turbulent eddy viscosity coefficient -q can be obtained from equations (XIII. 16
and 17)

rj = phl^ = pU^KZ. (XIII.20)

In the lowest bottom layers it will at first increase linearly with distance from the
bottom; but above a certain height it is generally assumed to remain a constant.

There are very few oceanic observations with which it would be possible to test this logarithmic
law for ocean currents above the sea bottom. This would require measurements at close intervals from
just above the bottom to a considerable height above it. The measurements made by Merz (Moller,
1928) in the southern entrance to the Dardanelles, which is sufficiently wide for the current to be un-
affected by the lateral boundaries, are probably suitable for this. Only the layers just above the bottom
need to be considered. Here the rather strongly scattered individual values of the three series of measure-
ments gave the following distribution:

Height above the bottom (m) . . 2 7 12 17 22 27
II (cm/sec) 0-3 2-8 4-6 5-5 6-5 7-2

These values follow a logarithmic law rather well and lead to the equation

It z

-= 5-75 log j--;z.

u^ 1-32

The representation of the observations by this equation is entirely satisfactory. It is of interest that in
spite of the certainly rather pronounced unevenness of the bottom (hence a large value for Cq)
the quantity Zq introduced above is apparently zero. This may be because the heights z above the bot-
tom are already heights above a "mean" sea bottom and in actual fact already represent z -1- Zq.
This dependence of velocity on height appears to apply only up to 25 m above the bottom. As
shown by observation the behaviour of u is then higher up completely different.

Current measurements near the sea bottom have been made by Mosby (1947) in order to study tur-
bulence and friction in the bottom layers. Using a special apparatus he has measured the direction
and intensity of the current in the Avaerstrommen (near Bergen, Norway) up to 2 m from the bottom
over a period of 3^ h; this gave the following mean vertical distribution of the horizontal velocity:

z (cm above the bottom) . . 25 50 75 100 125 150 200

M(cmsec-i) 16 23 27 29 31 31-7 32-5

These values can be represented rather well by the equation

^^ = 5-75 log ^^^.

It does not seem to be necessary to consider Zq in the formula. Later measurements (1949) did not show
such simple conditions; in the bottom layer (just above the sea bed) the velocity fell off very rapidly
to small values. The changes in the «-values with time at different heights above the bottom show clearly
the turbulence of the current; it appears to decrease only very slowly towards the bottom.

General Theory of Ocean Currents in a Homogeneous Sea 391

{b) Dissipation of Energy by Turbulence

The turbulent process mixes neighbouring water quanta; part of the energy is
deviated from the direction of the mean basic current, the water masses are flattened
out by vortices into thin layers and part of the energy is used up in this, which would
otherwise remain in the basic current. The magnitude of the energy dissipation by
turbulence can be calculated from the size of the shearing stress (XIII. 13). This shear-
ing force acts horizontally ; the relative movement of two water sheets one above the
other is dujdz. From this the work done by the turbulence (energy consumption by the
apparent friction "Scheinreibung") will be i? = rj(8uldzy. This is that work which must
be done in unit volume and unit time to maintain the turbulence against the velocity
gradient. (Schmidt, 1919).

In the example described above, in the Dardanelles, the velocity decreased from 27 m
down to 2 m above the bottom by 6-9 cm/sec. The mean velocity gradient was thus
{dujdz) = (1/362). The dissipation of energy per day amounted to 0-6677 ergs per cm^.
This appears rather small but over a longer period has an appreciable effect. If t^ =
100 cm~^ g sec~^ then the kinetic energy of a current of 20 cm/sec will be 200 erg/cm^
and this would be entirely absorbed by the turbulence in about 3 days if not continu-
ously renewed by other forces.

That the turbulence is dependent on the stratification in the medium is apparent
from the following considerations (Ekman, 1906; Schmidt, 1917; Pettersson, 1930,
1935). In the presence of stable stratification the mixing process is affected by the double
work required to lift the lower heavier water masses against gravity and to lower the
upper lighter ones against buoyancy forces. This hinders mixing and if the density
differences become large enough the stability of the water stratification reaches so
high a value that turbulence cannot act against it and may cease entirely. In subtropical
oceanic regions cases occur in the tropospheric deeper currents in which a thin layer
of highly saline water embedded between two layers of low-saline water can spread
over thousands of miles without being absorbed in the layers above and below by
mixing. The strong stabihty of the vertical stratification of the water masses completely
prevents mixing. An example of this behaviour of the subtropical intrusions of highly
saline water has been given in Pt. I, p. 169, Fig. 73 and the reader is referred to the
discussion at that place.

The conditions under which the work expended in the vertical displacement of
water elements by turbulence becomes so large that the turbulence is completely
suppressed can be found by comparison of the energy dissipation by turbulence and
the lifting work done against gravity by mixing. The buoyancy force per unit
time and unit mass for a density gradient dpjdz is given by g{/l p)(8pl8z).* The vertical
disturbance velocity u'' according to the previous discussion can also be put propor-
tional to I{dujdz). From (XIII. 16) and taking into account that for an equilization of
the density differences (temperature and salinity), iq must be replaced by the exchange
coefficients for the material properties of the water ^4^ (pt. I, p. 103), it follows that the
work done against gravity in unit volume and unit time is g(AJ p)(Spjdz). The work

* The symbol 8 should indicate the necessary consideration of the changes in density due to adia-
batic temperature changes.

392

General Theory of Ocean Currents in a Homogeneous Sea

done by the turbulent motion in unit volume is, however, rj(8uldzy. The condition
for the decrease of the turbulence in the disordered flov^ and its transformation into
an ordered flow is thus that the dimensionless stratification quantity

(glp)(Spl8z)

(duldzf

(XIII.21)

In earlier investigations it has mostly been assumed that rj and A are numerically
equal, i.e. that the mechanism of mixing of a material property is identical with that
of the impulse or momentum transport. Then 17 would be equal to A, and since the
stabiUty of the stratification would be given by (l/p)(8p/ez) = E (pt. I, p. 196), the
condition for the suppression of the turbulence would be

;^> 1.

(XIII.22)

(duldz)'

The expression on the left-hand side has been denoted the Richardson number Ri. The
upper limit at which all turbulent motion is extinguished is thus given by Ri = 1 ; how-
ever, in reality smaller values are sufficient. Referring to the latter statement, theoretical
and experimental investigations of Taylor (1931) and Goldstein on small oscillations
in a stratified flow with a linear decrease in velocity have shown that the limit can be
expected at Ri = 0-25 or |.

In oceanography it has usually been found (see pt. I, p. 104) that the ratio rj-.A is
of the order of 5 to 20. In the equatorial regions of the Atlantic Ocean in the density
transition layer (thermocline) dpjdz is of the order of 3 to 9 X 10-* for a 20 m height
interval. The decrease in velocity du/dz should be between 5 and 10 cm/sec for every
20 m, so that Ri must be between 6 and 69 (Defant, 1936c, p. 296 and 363). It is
clear that these figures are sufficiently high to prevent the occurrence of turbulence in
the tropospheric deeper currents, as has been found by observation.

Observations at two stations in the Baltic for which there was almost no turbulence
to be observed in the transition layer gave according to Gustafson and Kullenberg
(1936) Ri-numbers of 0-59 and 0-95 which are in accord with the hmiting values given
by Taylor. Detailed measurements have been made by Jacobsen (1913, 1918) at
Schultz's Grund (Kattegat) and in the Randersfjord, which are very suitable for
answering the question under consideration. Table 1 17 give as summary ofall the values
derived from these measurements.

Table 117. Turbulence and Ri-numbers at Schultz's Grund {according to

Jacobsen)

Depth
(m)

duldz

(cm sec"^

cm~")

Salinity
gradient
per cm

1 dp

P dz

(gcm-

^ sec"^)

2-5

10 X 10-3

10 X 10-«

7-5 X 10-^

0-3

7-1

111

11-2

3-1

0-4

3-8

7-7

7-5

28-5

2-7

018

5-9

14-9

100

600

2-2

005

10-2

43-5

12-5

140

1050

1-9

004

28-6

47-6

150

111

82-5

3-8

0-2

1250

200

General Theory of Ocean Currents in a Homogeneous Sea 393

At all depths -q has about 10 times the magnitude of the exchange coefficient A
determined from salinity measurements made at the same time. The quotient -qjA is
almost always larger than the Ri-number and therefore according to the above con-
dition is not compatible with turbulence. The Ri-numbers, which vary between 2-6
and 125, are so high that also according to this criterion a turbulent flow can hardly be
present. However, the measurements indicated still a small, though very weak,
turbulence with a frictional coefficient between 1-9 and 3-8 g cm~^ sec~^. According
to these investigations, other factors seem also to be involved in the appearance and
maintenance of turbulence (close distance to a solid boundary or the presence of an
intermediate layer between the otherwise almost homogeneous water masses above and
below).

{d) Turbulence and Mixing in the Sea; Statistical Theory of Turbulence

The modern hydrodynamic approach to ocean currents has led increasingly to the
view that the turbulence of the ocean currents, which finds its visible expression in the
oceanic mixing processes, is the basic cause of a number of oceanic phenomena.
Oceanography has mostly been concerned solely with the effects of turbulence and
mixing on oceanic phenomena; only recently has interest been directed also towards
the nature of oceanic turbulence and one has asked the important question : of what
kind is this nature ? In laminar flow the velocity can be represented by a simple function
of position and time. In turbulent flow the mean velocity, which again can be repre-
sented by a simple function of this sort, is superimposed on an additional, irregularly
varying turbulent velocity component that changes with both time and space. The
sharp distinction between the two types of flow is shown by experimental investigations
which indicate that a discontinuous transition from laminar to turbulent flow occurs
when a dimensionless quantity, the Reynolds number, exceeds a critical value, the
magnitude of which is about 1000. The form of the Reynolds number indicates the
cause of this basically different behaviour of the two types of flow. The Reynolds
number is given by R = plJL\r], where p is the density, U and L are values for the
velocity and the hnear dimension which are characteristic for the structure of the
particular current under consideration; r] is the eddy viscosity coefficient (frictional
coefficient). It is clear that the current will be turbulent when the momentum (impulse)
of the flow pU or the distance L passed through are large; it will be laminar if the
viscosity is large. The viscosity is a force carrying neighbouring elements of the
medium along the same path. Therefore, it is obvious that large viscosities will have a
tendency to smooth the course of the flow. The empirical fact that the current tends to
change to turbulent flow even with very small disturbances — i.e. that the laminar
flow is unstable — shows that the turbulent flow has in a certain sense to be regarded
as the natural form of motion of media with low viscosity. The Helmholtz vortex-laws
of classical hydrodynamics show that a vorticity-free current cannot develop vortices
spontaneously. Thus no turbulence can occur in it by itself. It can only be produced
inside the fluid by friction at solid surfaces, or by similar processes through the forma-
tion of vortices at the boundary of the liquid. Once formed it will spread out in the
fluid. This is, however, not the case which we meet in the open sea remote from the sea
bottom and from the coasts. The ocean currents here usually have a considerable vortex-
intensity from the beginning, i.e. from their formation; it is their further distribution

394 General Theory of Ocean Currents in a Homogeneous Sea

on vortices of smaller dimensions that has to be regarded as the turbulence of the
current. The origin of the oceanic turbulence must thus be traced back to the con-
ditions of formation of the ocean current, and this can definitely be considered to have
been done, since the conditions which prevail initially during the formation of the
current are certainly scarcely of the type that could be described by simple functions
of the velocity distribution. On the contrary, everything indicates that during the forma-
tion of a current due to the complicated distribution of the shearing stresses of the
winds, the ocean current looks right from the beginning rather confused in vertical
and horizontal direction, so that a priori there is a very large probability that in the
future the resulting current will attain a form which will fall within the general concept
of turbulence.

Turbulence is not a form of motion that can maintain itself indefinitely. The kinetic
energy of the current is continuously converted by the molecular viscosity into heat.
If the current is not continuously supplied with fresh energy, it must in time die away.
In the ocean, the currents are continually supplied with energy by the tangential
shearing forces of the winds so that here steady turbulent currents are possible. This is
of particular importance to the nature of ocean currents which are recognized as
essentially quasi-stationary phenomena by observations.

Turbulence and mixing in vertical direction and also lateral turbulence of the ocean
currents were already discussed in § III dand e of Pt. I of this volume. Lateral mixing
is on a much larger scale than the vertical ; the turbulence elements are of considerably
larger dimension, so that the eddy viscosity and eddy diffusion coefficients are very
large. The ratio of vertical to lateral mixing coefficients is of the order of 10^ to 10'.
It can be shown both experimentally and by observation that there is a "continuous
spectrum" of mixing and turbulence coefficients extending from the molecular vis-
cosity coefficients to values for the eddy conductivity of 10^^ (one billion) or more
(Richardson, 1926).

In a turbulent current where u is the velocity at a certain point and varies with time,
the basic velocity is defined as (time interval 7") :

1 r
U=^\ u(t)dt

and further the supplementary turbulent velocity as u' = u(t) — U, whereby

1 r

- u'(t) dt = 0,

the intensity of the turbulence is given by 7 = 1/\/{(m')-} and its kinetic energy by
E = ip(M')^.* These quantities characterizing the turbulence of the flow depend of
course on the length of the time-interval T, and in fact a sufficiently large value for
T has to be selected or these quantities lose their meaning altogether. In laboratory
experiments in wind tunnels this requirement can always be closely approached, but
whether this is also the case for oceanic water masses is difficult to judge. If T is less
than a few hours then the »'(0-values will include terms for the small-scale turbulence
such as local mixing, while the basic velocity U will include the long-periodic variations

The bar above a quantity indicates its mean value taken over the time-interval T.

General Theory of Ocean Currents in a Homogeneous Sea 395

in the velocity such as the tidal currents and the annual changes in u' . If T is selected
with a value of about a month the tidal currents will also be included in the value of
u'{t). If ris chosen for 10 years or more, the seasonal changes will also be included in
11 and only the secular changes will remain in U. From this it can be understood that,
in nature, motions in water masses as they appear in the ocean will be much more
complicated than, for example, in an experimentally controlled wind tunnel or a
water channel. Every size and all different velocities of the turbulent vortices can be
expected to occur in oceanic turbulence, and it is not easy to distinguish between the
basic velocity and the additional turbulent velocity. These difficulties occurring with
turbulent phenomena of the ocean and atmosphere seem to be fundamentally connected
with the nature of turbulence.
In dealing with mixing processes in the ocean, the simple relationship

ds d"s

Jt ^ ^8z2

has usually been used, where S{z, t) is the concentration of the diffusing substance and
K denotes the mixing coefficient (eddy diffusivity, eddy conductivity), [cm^ sec~^].
This is termed the "Fickian diffusion equation" (see Pt. I, pp. 95 and 104). It is derived
by analogy with molecular processes for the larger-scale processes in turbulent currents
using simplifying assumptions on the internal nature of turbulence; it does not accord
fully with more recent data, and especially not with the fact that the larger the mixing
coefficient becomes, the larger the scale of the phenomena under consideration, i.e.
with the existence of a continuous spectrum of the diffusion coefficient.

With molecular diffusion, as described by the Fickian equation, the movement of
each molecule is independent of that of a neighbouring one. In contrast to this, how-
ever, in a turbulent current, adjacent elements have increasingly similar turbulent
velocities, and in fact the more there are the smaller the distance from each other. The
reason for this is easily understood when the behaviour and the effect of the turbulent
vortices of all sizes are studied altogether in detail. The distance between two initially
adjacent elements is altered only by the smallest vortices ; the effects of the larger
vortices cause no significant change in distance, since they give rise only to a simple
transport of these elements. If, however, the distance between two elements becomes
larger, the effect of the larger vortices is added to that of the smaller ones so that as the
distance between them increases the diffusion effect due to the larger-size vortices
becomes more and more involved.

The most important independent variable cannot be, as in molecular diffusion
processes, the position of an element, but the distance from its neighbouring element.
This requires that the concentration of a diffusing substance is only a function of the
mutual separation of the particles inside this substance and not a function of the posi-
tion only.

Richardson first showed this difference as compared with molecular diffusion and
further investigations have then been carried out to account for this circumstance
(Witting, 1933; Sverdrup, 1946; Proudman, 1948). The theory that the concentra-
tion of a diffusing substance is not a function of the position of the element which it
occupies, but rather of its distance / from the adjacent element leads to the conclusion

396

General Theory of Ocean Currents in a Homogeneous Sea

that the diffusion coefficient i^ is a function of the neighbour-distance / and is given
by the equation

F{1)=^

(/i - kf
2t '

(XIII.24)

where /q is the distance between the elements which are at the same distance in the
turbulent current at time / = 0, while / is the distance at time /. F can be determined
from experimental series-measurements from the values for / and this allows a de-
cision as to whether the Fick or the Richardson concept of the internal nature of the
turbulence fits the observed data; since according to the Fickian theory F must be
independent on / (see also, Ichve, 1950). All the observations made (Richardson,
1926; Witting, 1933; Stommel, 1949; Hanzawa, 1953; Inoue, 1952) show that F
is in fact strongly dependent on / and that there exists a definite relationship between

them of the special form

F(l) = f/4/3. (XIII.25)

Figure 167a shows a summary of observed data and it is easily seen that the assump
tion of a 4/3 power seems to be fully justified.

10"

o A

10'°

/ /

/ ^

10^

' A/

/ V

10^

/•//I/

/ ■/

V /

10^

/ /

7 /

/ ^

106

^ /

/ /

10^

l/ /

' /

10^
105

/ /

^ /

V +

V /

/ y

V >

102

_/^

/ .

A /

/ A/

10
0

/ ^

/ / / /

' 1

10"'

•

/ / A /

10"' 0 10 10^ 10^ lO" 10^ 10^ 10^ 10^

Fig. 167a. The relation /^(/) = e/*'^ according to observations (logarithmic scale): points,

values of Richardson from the atmosphere; crosses, values of Stommel (Blaimore, Bermuda

and Woods Hole); triangles, values of Hanzawa.

Equation (XIII.25) which has been found inductively has been given a sound theor-
etical basis by closer study of the rate of the energy decrease due to turbulent mixing
of the large-scale motion. This method of investigation was first introduced by Kol-
MOGOROFF (1941) and after some intermediate work Weiszacker (1948) and Heisen-
BERG (1948) have brought this statistical theory of turbulence to a certain degree of

General Theory of Ocean Currents in a Homogeneous Sea 397

completion. This theory leads to the same 4/3-power law for the turbulent exchange
coefficient which was previously derived from observations. With some modifications
this theory can be applied to large-scale processes occurring with oceanic currents,
and offers the possibility of obtaining a picture of the spectral distribution of energy
in oceanic turbulence. It is thus of a considerable interest for oceanography.

The semi-permanent wind systems such as the trade winds, the prevaihng westerlies
of temperate latitudes, and furthermore, the aperiodic air currents of the extra
tropical pressure disturbances, give rise to large-scale movements in the surface layers
of the ocean due to the shearing stresses acting on the sea surface. Thereby, these
shearing stresses tend to increase the kinetic energy of the currents produced. However
the mean kinetic energy of the ocean currents remains largely constant (quasi-
stationary conditions) so that finally as much energy is dissipated in heat as is gained
by the work done by the shearing stress of the wind. Ocean currents which initially
show large-scale turbulence tend to break up into vortices which subsequently
degenerate into smaller and smallest vortices. This proceeds until finally the smallest
vortices are formed, which are so small that their energy is converted in irreversible
processes by molecular viscosity into heat energy. An exact dynamic explanation of the
reasons why the large ocean currents break up into turbulent currents, with more or
less large vortices of widely varying size, has not yet been given. However, the em-
pirical facts of their existence have been shown by synoptic surveys, for instance, in the
more recent Gulf Stream investigations.

A complete spectrum of vortex sizes certainly exists. This spectrum is necessary for
the dispersion of the kinetic energy of the ocean currents continuously supplied by the
shearing forces of the wind. In practical oceanography it has long been recognized
that the concept of the mean velocity of the oceanic currents is rather dependent on
the length of the time interval over which its value was determined. The same applies
for space-means of the current intensity. This leads to the expectation that the mag-
nitude of the turbulent coefficients also depends fully on what kind of evaluation of the
mean has been used. The concept of a turbulence coefficient is absolutely meaningless
if the way in which the mean was found is not specified. This can be seen already from
the greater magnitude of the turbulence coefficients the greater the dimensions of the
movements under consideration; a fact which could not be explained in earher work.

The Weiszacker-Heisenberg statistical theory provides information on the fre-
quency distribution of the energy in different size-intervals of turbulent vortices, on
the way in which the mean velocity depends on the type of mean taken, and lastly on
the dependence of the turbulence coefficients on the type of mean taken.

If Ln is the side of a square over which the nih. mean is taken then according to
Weiszacker the spectral law is, for the turbulent velocity distribution :

z7„ proportional to L)^^,

for the turbulence coefficient:

7j„ proportional to L,^^
and for the turbulent energy distribution:

En proportional to L^J^.

398 General Theory of Ocean Currents in a Homogeneous Sea

Weiszacker took a discrete velocity spectrum as the basis of his theory, Heisenberg
chose a continuous velocity distribution and provided an elegant mathematical proof
(in this connection see also Ichve, 1951).

The principal result of the theory, as far as it concerns the exchange coefficients of
turbulent motion, is in complete agreement with the 4/3 power law derived from
observed data. The more recent statistical theory of turbulence can give a better
description of actual conditions in nature than the classical Fickian theory. In par-
ticular, the theory gives an explanation for the large differences in size between the
turbulence coefficients for small- and large-scale motion, for which there was no ex-
planation in earlier time. For small-scale oceanic phenomena the values found
for the diiTusion coefficient t] are on the average about 50-100 cm^ sec~^. For large-
scale ocean currents, on the other hand, the values were between 10^ and 10^ cm^ sec"^.
The ratio between these is about 5 X 10^ to lO**. For small-scale processes L can be
taken as about 50 m and for large-scale currents as about 1000 km. The ratio of the
L-values is 2 x 10* and for the T^-values should be according to the theory about
5-4 X 10^. The agreement with the values derived from observations is rather good.

The question could also be raised, how far the assumptions made by the theory are
justified in oceanic conditions. Stommel (1949) has closely examined this question.
Not all the sources for turbulence in the ocean are due to air currents, a part is cer-
tainly due to the thermo-haline structure of the ocean currents the dependence of
which, of course, on solar radiation and evaporation is known. The assumption of a
continuous series of vortex sizes with horizontal isotropy can hardly be valid for the
large oceanic vortices ; it can be postulated as a first approximation only when they are
of smaller dimensions, i.e. for the genuine turbulent vortices of oceanic currents. The
changes which should be introduced for oceanic conditions involve the dividing of the
vortex sizes into two parts : an anisotropic one, including all the kinematically dissimi-
lar, large-scale horizontal movements, and an isotropic part, including all the kine-
matically, similar-to-each-other, turbulent vortices. The latter part only appears after
a certain nth averaging process. The first part is thus essentially concerned with the
advection of different water types. The exchange is only involved in the second, and
the statistical theory of turbulence should be fully applicable here. However, in spite
of these changes in many of the assumptions the basic idea of the theory remains and
offers a solid basis for the study of dynamic conditions of the ocean currents.

4. Steady Currents in a Homogeneous Ocean under the Action of External Forces

(a) Introduction

The first ideas about the effect of friction on the movement of water masses were
based on the assumption that it arose from the roughness of the bottom surface
(gliding friction). The frictional force was thus given, as already shown on p. 317, by

R = -KpV.

GuLDBERG and MoHN (1876) using this principle for atmospheric flow presented a
diagram of the forces necessary for a steady motion. It can also be applied to water
movements in shallow ocean currents for which the frictional effects of the bottom
act throughout the entire water column. In that case the resultant of Coriolis force
and frictional force must balance the gradient force. The direction of the current is

General Theory of Ocean Currents in a Homogeneous Sea 399

now no longer parallel to the isobars but is deflected at an angle proportional to k.
On the right-hand side of the equations of motion (XIII. 1) the components for the
frictional force —ku and —kv have to be added. Multiplying the first equation by u
and the second by v and adding, gives

1 dW 1 dp

2 dt- p dt

For the movement of a water element along an isobar (dpldt = 0) this equation gives

K= VQe-'<K

The velocity of the current which is acted upon by Coriolis force and friction, usually
decreases until it vanishes. The value l//c gives the time needed by the bottom friction
to reduce the velocity by a factor of 2-72. For currents in shallow waters k is of the
order of 10"*^ to 10~'^ sec~^, so that the velocity of the water movement will fall to a
tenth between 2 and 25 days.

The Guldberg-Mohn frictional principle makes no allowance for the fact that a
turbulent flow is aff'ected also from above by mass exchange with the layers above it,
in addition to the eff'ect of the bottom surface which affects the flow from below.
Sandstrom (1910) has taken this circumstance into account by assuming that the
frictional force does not exactly oppose the current, but its vector deviates by a small
angle to the left of the current direction (or the force acts backwards and to the right
of the current).

Also this frictional principle can only be considered as a makeshift and gives ac-
ceptable results only for currents in very shallow waters. If all the factors involved in
the formation and maintenance of the ocean currents are to be taken into account it
is necessary to return to the hydrodynamic equations of motion in the form given in
(X.16). Besides friction, there must also be taken into account the effect of the Coriolis
force and as current producing factors, especially the tangential pressure of the
wind on the sea surface, the pressure gradient and gravity. For horizontal water trans-
ports, i.e. along the gravitational level surfaces, gravity is less important as an im-
pelling force. If only the wind stress, the Coriolis force and friction are acting, the
current will be a pure drift current; if, however, gradient force, Coriolis force and
friction are the decisive factors, it will be a pure gradient current. The following section
is concerned with these two basic forms of water movement.

The fundamental work in this direction is due almost entirely to Ekman (1905, 1906,
1922) who first gave a strict mathematical form to the eff"ects of the Coriohs
force and friction in the theory of the ocean currents in a homogeneous sea. The great
significance of these two forces for the generation of drift currents had already been
recognized and demonstrated by means of observational data by Nansen (1902,
1905). These investigations opened a first way to the development of a complete
theory of ocean currents.

{b) Pure Drift Currents

A pure drift current is a result of the wind stress acting on the surface of the sea.
This stress is produced either by friction of the air passing over the water, or by the
pressure effect of the wind on waves which transfers part of the momentum of the

400 General Theory of Ocean Currents in a Homogeneous Sea

wind to the water. Both effects usually act in the same direction and can be com-
bined as a single tangential force. If there is no pressure gradient within the water mass
the surface of the sea must be level {(dpidx) = (dpidy) = 0}. With this the condition
of an infinite extent of the ocean is basically connected, since otherwise the currents
produced will give rise to a piling up of water at the coast lines which will tend to form
gradient currents. Such currents will, however, for the moment be disregarded here.
In the case of a steady acceleration-free horizontal current {{dujdt = {dvjdt) = 0
and vt' = 0} and for constant frictional coefficients the equations of motion (X.16)
will take the form (/= 2w sin ^, z positive downwards):

d'^u 8^v

pfv + V-f:2-^ and -pfu + v^2 = ^- (XIII.23)

Multiplying the second equation by / = \/—\ and adding to the first gives

1^2 (" + iv) = — (m + iv). (XIII.23fl)

For practically unlimited ocean depths the general solution can be taken in the form

u + iv =^ A e-Ci+'X-^/'D), (XIII.24)

where

\l\fpj ~ ^ \J [poj sin cj^J-

The boundary condition that the velocity of the drift current vanishes for large depths
(z = co) is already satisfied by (XIII.24). At the surface of the sea (z = 0), a wind in
the direction of the positive j'-axis will give rise to a shearing stress T, which can be
represented by the relation

^(" + iv)

for r = 0. The solution then takes the form

M + /y = (1 + /) J^ ^-(i+»>/z)^ (XIII.25)

Ittt]

From this the two velocity components of the drift current are then obtained
M = Fo e--"D cos 1^45° - ^-) and v = V^e--''^ sin (45° - ^-) (XIII.26)

D = TT

with

" \/(2)Dpoj sm(f> \J \pco sm

At the sea surface the water in a pure drift current moves with a velocity V^ in a
direction 45° cum sole from the wind direction. At increasing depth the angle of de-
flection increases while at the same time the velocity of the current rapidly decreases.
At a depth D the deflection will amount to a full 180° and the velocity will have fallen

General Theory of Ocean Currents in a Homogeneous Sea 401

to e~'" = 1/23 of the surface value. This velocity is already so small that by com-
parison with the surface value it can usually be neglected. The depth D can therefore
be taken as a measure of the depth of penetration into the sea of a v/ind-generated
ocean current on the rotating Earth. It can in general also be taken as a measure of
how far downwards the effect of a steadily flowing horizontal layer penetrates into the
adjacent water masses. It was termed by Ekman the ''frictional depth'"; for drift
currents the additional word "upper" is used in order to indicate that here solely
conditions in the top-layer of the ocean are dealt with.

According to equation (XIII.26) D can also be taken as a measure of the internal
turbulent friction. It should be noted that the shearing stress T is not involved in the
equation relating D and rj ; this could be interpreted to mean that the vertical thickness
of the drift current should be independent of the wind intensity producing it and
maintaining it against friction. This apparent contradiction is clarified by consider-
ing that the frictional coefficient increases with increasing wind strength as does also
the frictional depth D.

Figure 168, according to Ekman, shows the vertical structure of a pure drift current;
the arrows projecting from the central column which are also shown in a projection

Fig. 168. Vertical structure of a pure drift current (according to Ekman).

402

General Theory of Ocean Currents in a Homogeneous Sea

s °'

I 0-2
S 0-3
§ 04

i 0-5

° 06

I 07

E 08

£ 09

.^ 10

y II

f. 1-2

-01 0 0-1 0-2 0-3 04 Q5 0€ 07 08 0 9 10
Velocity relative to the surfoce

Fig. 169. Vertical current distribution in a pure drift current: (a) in the direction of the
surface current; (b) normal to the direction of the surface current.

I -4

Fig. 170. Vertical structure in drift currents for an ocean depth J nearly equal or smaller than

the upper frictional depth Z) (10 small circles indicate on each curve the end-points of the

velocity vectors for the depth 0-0, 01, 0-2 ^ and so on until 0-9 d. The dashed curve at 1-25 D

refers to d = 2-5 D, the remaining part coincides with the curve for 1 -25 D).

M = \ (m + iv) dz =

J 00

General Theory of Ocean Currents in a Homogeneous Sea 403

on a horizontal plane, give a representation of the direction and strength of the current
at the surface and at equidistant levels O-ID, 0-2i), etc. The arrovi^ at the peak of the
vertical Hne represents the direction of the wind. The arrow-heads he on a doubly
curved spiral and the end-points of the vectors on the horizontal plane lie on a logar-
ithmic spiral (Ekman spiral). Referring the current components to the direction of the
current at the surface and at right angles to it the diagram pictured in Fig. 169 is
obtained, which allows one immediately to judge whether the observed vertical dis-
tribution of the current carries the character of a drift current.

Equation (XIII.26) shows further that the sea surface velocity increases in propor-
tion to the shearing stress T but in inverse proportion to the frictional depth D.
This is reasonable since, for equal Tthe more water that is set in motion, the smaller
must the velocity of the drift current be, i.e. the greater the depth D. The total drift
current transport per unit area of the sea surface is given by

that is

M^ = (Tjf) and My = 0.

The total water transport due to a drift current occurs perpendicular cum sole to the
direction of the shearing stress of the wind producing it and since rj is not involved
it is independent of the assumption concerning the effects of eddy viscosity. For an
arbitrarily chosen co-ordinate system with shearing stresses T^ and Ty in the x- and
>Mlirections, the water transports in these directions will be

M^ = ^ and My= - j. (XIII.27)

Finite water depth. When the depth of the water is about of the same order as D it
has a noticeable effect on the drift current. For a depth d the e-functions in the solu-
tion will be replaced by hyperbolic functions. At the sea bottom (z = d) u = 0 and
V — 0 are assumed as boundary conditions indicating "adhering" ("Haften") of the
water on the underlaying surface. It is apparent from this solution and follows also
from Fig. 1 68 that as long as the depth of water is greater than the frictional depth D
the vertical distribution of the drift current will be unaffected, since the water layers
below the frictional depth have an insignificant share in the drift current. When, how-
ever, the water depth d becomes smaller than D, the effect of the bottom will be of
more influence the shallower the sea. Figure 170 shows the vertical current structure
for depths d = 1-25D, 0-50D, 0-25 D and 0-lD. The thin dotted curve near the origin
of the co-ordinate system for the curve ^ = 1-25 D shows the deviation towards the
curve for an infinitely large depth; thus in practice there is no significant difference
between them. The angle of deflection decreases rapidly with the depth of the water
and at very small depths, approximately from about d <0-\D, the movement shows
almost no effect of the Earth rotation.

Other frictional assumptions. In addition, Ekman has given a solution for the case
where the frictional coefficient is proportional, not to the difference in velocity be-
tween two adjacent layers, but rather to its square. This gives essentially the same
results as for a constant -q ; the angle of deflection of the sea surface current is now

404 General Theory of Ocean Currents in a Homogeneous Sea

49-1° and the current dies away at the finite depth of \-25D. It should be pointed out
that the relationship between T, D and Vq are somewhat different. The total transport
for the quadratic frictional law is, however, also given by (XIII.27) and is thus inde-
pendent of the frictional assumption. This can also be shown by strict mathematical
treatment. For a variable -q the expression

d\u, v)

in equation (XIII.23) is replaced by

d I d(u, v)
8z [ ^ -d^-

see p. 319. Integrating this equation from z = 0 to z = oo or respectively down to a
depth at which the drift current can no longer be detected, and considering that the
shearing stress is present only at the sea surface, then with the help of equation (XIII. 13)
relationships are obtained which are identical with (XIII.27). These, however, were
derived for a constant rj. It could possibly be expected that during the transfer of the
turbulent wind momentum to the water masses at the sea surface the two horizontal
components of the shearing stress (in the direction of the wind and at right angles to
it) would be governed by different turbulent coefficients. An extension of the Ekman
theory along such lines has been given by Ertel (1937). It leads to deflection angles
different from 45° while the vertical current structure becomes a deformed spiral.

Another principle applicable both to the wind stress at the sea surface and to the
friction at the bottom has been developed by Jeffreys (1923), In conformity with
turbulence theory he assumed that at both the sea surface and at the bottom, "gliding"
of the water masses occurs in which the friction is assumed proportional to the square
of the velocity differences. The boundary condition at the sea bottom is taken as

and at the sea surface as

- 7] —^ = Kp{u\ v^)

^("'^) V '2 '2^

where p' is the density of the air and u' and v' are the velocity components of the wind
relative to the water movement at the sea surface (see p. 317, equation (X.9).)

The more recent results of research in turbulence also show that in the vicinity of
boundary surfaces the assumption of a constant frictional coefficient leads to current
distributions which do not accord with the observed facts. This makes it necessary to
introduce turbulent coefficients, wliich vary with the distance from the solid boundary.
That such a method leads to results satisfactorily explaining the observed features has
been shown by an investigation of Fjelstad (1929) using observations made by Sver-
drup on a drift current over the North Siberian Shelf, where there was a strong increase
of the frictional coefficient from the bottom to the surface. He succeeded in deriving
a functional relationship for these coefficients of the form

fZ+ €\8/*

1 =^ Vo '

General Theory of Ocean Currents in a Homogeneous Sea

405

and was then able to obtain a solution for the corresponding equations of motion
Fig. 171 presents the vertical distribution of the frictional coefficient as well as of the
theoretical current structure, both for a constant frictional coefficient and for a coeffi-
cient varying with depth, according to a summary made by Thorade (1931). The
observed current values are indicated by crosses. There remains no doubt that agree-
ment with the observed data is obtainable only by using coefficients variable with
depth.

(o)

(b)

/
/

/ 1

/
/

+Onn

" /

y/-

ilOm
/l2m

-*r?Oni ^

■^I5m

•0 100 200 300 400

Bottom

Fig. 171. (a) Vertical distribution of the turbulent coefficient at a station of the North

Siberian shelf, (b) Current diagrams: --0--0--, theoretical distribution for a constant

frictional coefficient ; — o — o — , theoretical distribution for a frictional coefficient as in (a);

+ + + + + + +, the observed values according to Sverdrup.

The application of the modern theory for a turbulent flow to drift currents will be
discussed later together with its application to gradient currents (see p. 311).

Effect of stratification. Assuming a horizontal and stratified sea with a normal
density increase with depth, then only minor deviations occur as compared with the
case for a homogeneous sea (Defant, 1927). However, essentially different conditions
appear for sudden vertical density changes (boundary surfaces between different water
masses). Here the stratification affects especially the frictional coefficient, which inside
the flow of each more or less homogeneous water mass may remain approximately
constant and relatively large but may fall almost to zero inside the density transition
layer (thermocline). The effect of the wind is thus confined essentially to the top layer
and the drift current in this is transmitted only very slowly to the lower water mass
across the transition layer. As a boundary condition at the side beneath the top layer
it must be assumed, since the water here meets almost no resistance, that there is
perfect "gliding" and the drift current in the top layer will thus be different from that
over a solid surface. If ^is the thickness of the top layer (z=d) this boundary condition
is given by

7 - = 0 for (z^d).

cz ^ ^

Solutions of this sort have been discussed in greater detail by Nomitsu (1933). The
shallower the layer of water in motion the stronger is the current produced by the
wind and the larger the angle of deflection; a result which is exactly opposite to that

406 General Theory of Ocean Currents in a Homogeneous Sea

of the previous case of "adhering" ("Haften") at the sea bottom. For a small thickness
an almost geostrophic current is obtained. As the thickness of the layer increases,
the structure of the current will of course approach that of the Ekman spiral.

Drift currents in normal form are seldom found to occur in the sea, since the water
transport connected with such currents will give rise to piling up of water at coast
lines ("Anstau") leading to inclination of the sea surface. In a homogeneous sea the
pressure differences produced in this way would extend their influence down to the
sea bottom; if there were no frictional effects a geostrophic current would be generated
from the sea surface down to the sea bottom. However, friction at the bottom gives
rise to disturbances which are of considerable importance for oceanic currents.

The equation of motion (X.16) for a steady current will be of the form

\ dp 7] 8^u ^ , ^ \ dp 7] 8^v

fv- - -/ + i — = 0 and - fu - - -^ + - 5-0 = 0. (XIII.28)

p ox p oz^ p oy p cz^

Replacing the pressure gradient by the slope ^ of the sea surface (equation (XIII.2),
p. 383) and assuming that there is a pressure gradient only along the j^-axis
(dpidx) = 0, then, according to (XIII.5), the geostrophic current will flow in the
direction of the positive x-axis and its velocity will be

g S^

Considering this in the equations (XIII.28) they can be compressed in the same way as
for a drift current into

- gZ-2 («+'■") - '/(" + 'i') + //t/ = 0. (XIII.30)

To this equation add the following boundary conditions:

(1) no wind at the sea surface, that is

Su cv ^
forz = 0: =-=0

cz oz

and

(2) at the sea bottom "adhering" occurs ("Haften")

for z = d: u — v = 0,

The solution given by Ekman for (XIII.30) is

cosh(l - i){7rlD)z

u -\- iv = U

^ cosh{\ +i){7T I D)z

^U(l-<f>'h i>P),

whereby

cosh (7rlD)(d + z) cos (-^iDXd + z) + cosh (nlD)id — z) cos (nlOXd — z),

<f> =

cosh 27T(dlD) + cos 27r(^/Z))

(XIII.31)

General Theory of Ocean Currents in a Homogeneous Sea

407

for ^ the functions cosh and cos are replaced m the numerator by the complementary
functions sinh and sin. Thus

u = {\ -4>)U and y = 0C/, (XIII.32)

D denotes again the frictional depth to which now, since it refers to the sea bottom, is
supplemented the additional word "lower". The functions (/> and </» determine the
vertical velocity distribution of the gradient current. For z = d one obtains ^ = 1
and </» = 0 as required by the boundary conditions. Its further course is best shown by
evaluating the equations for different values of dID. Figure 172 presents as an example

0 0-2

0-4

' 1
if.Z

D' 2

1
''-1

D '

D z\

^^tt^'''^

Z?=4

Fig. 172. Vertical distribution of the velocity components u and v for different values of h\D

(for values hlD = if the course of the M-component coincides with the straight line 1-0;

the t;-component approaches rapidly the straight line 00).

the vertical distribution of the two components for the d\D = 1-5, 0-5 and 0-25. In
the curves for depths somewhat greater than the frictional depth the course of both
components is the same. Until a depth above the bottom is reached corresponding to
the frictional depth u increases rapidly and reaches here the value of the geostrophic
current U. The M-component increases a little further but then reverts to the tZ-value
and remains then almost constant. The r-component (in the direction of the pressure
gradient) rapidly reaches a maximum not far above the bottom, then falls almost to
zero and oscillates with decreasing amplitude around the zero value. For depths
d > 1-5 D the structure of the pure gradient current has the form shown in
Fig. 173; this is drawn in the same way as Fig. 168. At distances from the bottom
greater than D there is a practically uniform velocity at right angles cum sole to the
pressure gradient. This is the uniform deep current; it corresponds to the frictionless
geostrophic current. The bottom layer is governed by the bottom current, the velocity
of which decreases according to a logarithmic spiral down to the sea bottom. For
greater depths of the sea the only change in this structure is in the vertical thickness of
the deep current ; the bottom current always corresponds to the frictional depth D.
Since the deep current runs parallel to the topographic lines ("Niveaulinien") of

408

General Theory of Ocean Currents in a Homogeneous Sea

the sea surface it cannot contribute to the equalization of the sea surface slope. This
can only be accomplished by the bottom current which always has a component in the
direction of the pressure gradient, i.e. a transport of water from a higher to a lower
level. This component does the work required to overcome bottom friction.

Fig. 173. Vertical structure in a pure gradient current (according to Ekman).

The transports (current amounts) M^ and My of a gradient current can be calculated
by integration between 0 and d of equation (XIII. 32), after its multiplication by p.
In case of no bottom current the current component M^ would be Upd and it becomes
smaller due to the velocity decrease at the bottom. One obtains

M^ = Upd - U

Itt

For depths less than D the effect of bottom friction is noticeable throughout the entire
water layer, and the more so the smaller the ratio djD. The curves in Fig. 174 illustrate
the gradient current at depths 1-25 D, 0-5 D and 0-25 D. The angle of deflection

d=i-zsD

Fig. 174. Vertical structure of gradient currents for ocean depths d nearly equal or smaller
than the lower frictional depth D (for more detail see Fig. 172).

General Theory of Ocean Currents in a Homogeneous Sea 409

between the current and the gradient direction becomes smaller and smaller as the
sea becomes shallower; the effect of the Earth's rotation then becomes less important
than that of friction.

Other assumptions about friction. The Ekman theory assumes a constant frictional
coefficient. It has been used in this form in meteorology and provides an unobjec-
tionable explanation of the deflection of the wind direction to the right with increasing
height. However, it was found that the lowermost layers of the wind structure follow
different laws. These deviations can be attributed mainly to the assumption of a con-
stant frictional coefficient in the bottom layers being no longer valid. This fact
Ekman (1928) has taken into account by assuming in agreement with the observations
a current structure made up of a straight section OA, at A changing into a logarithmic
spiral over AB (Fig. 175). Thereby OB is thus the geostrophic wind in higher altitude.
The same conditions as for the surface wind must also apply to the oceanic bottom

Fig. 175. Vertical structure in a bottom current with a boundary layer above the bottom

(according to Ekman).

current, and it is already known from current measurements in moving waters and
from laboratory experiments that the vertical structure in these, apart from the devia-
tion due to the Coriolis force, is somewhat different from that of the Ekman spiral.
The velocity curve of Fig. 175 can therefore only be given a physical m.eaning by
assuming the presence of a boundary layer just above the bottom in which the velocity
changes approximately linearly, and without change in direction from zero at the
bottom to the value OA = Vg at its upper limit. The water mass present above this
lower boundary layer flows as though gliding over the bottom ; it is retarded only by
the slowly moving boundary layer. Ekman assumed a constant frictional coefficient
in each of the two layers and investigated the thickness of the boundary layer, the
decrease in velocity in it and the angle of deflection which would be able to prove the
validity of such a concept.

This concept can more or less accommodate the fact that the lowermost layer just
above the bottom has a special status, and that in practice the assumption of a constant
turbulent coefficient in the water masses above is quite justified. Modern hydrodynamic
fluid research approaches the whole problem from the point of view that the variation
of the frictional coefficient with distance from the solid underlaying surface changes
with its roughness, whereby the entire current structure takes on a different form. The
Prandtl theory (see especially 1942, p. 318) starts with the components of the shearing
stress T^ and Ty at the bottom. Taking r-positive upwards, furthermore a variable 17,
a pressure gradient in the direction of the positive j-axis and taking into account
equation (XIII. 13), then equations (XIII. 28) can be transformed into

410 General Theory of Ocean Currents in a Homogeneous Sea

T^=f\ pvdz and Ty=f\ p(U - u) dz. (XIII.33)

0 0

Here h denotes the lower frictional depth at which the deviations U — u and v from
the geostrophic current vanish. It can further be assumed that T at the bottom has
the same direction as the velocity at the bottom, so that

= tan a, (XIII.34)

Z= 0

where a is the angle between the direction of the resulting T and that of the uniform
deep current. These relationships form the basis of the vertical current structure of the
bottom current, but further extension of the calculation fails due to the still imperfect
knowledge of the laws of turbulent flow. However, by use of the above presented
basics for turbulent friction a rather good estimate of the vertical velocity profiles
to be expected can be obtained.

As a first approximation it can be assumed that in the vicinity of the bottom u
varies with the «th root of z

u = U

(a'

Further, near the bottom v=u tan a ; in order that v vanishes at a height z=/7i one has to
assume

y = M 1 1 — r I tan a.

Ill is smaller than h and must be chosen so that the current structure near the bottom is
in accordance with that shown by turbulence research (equation XIII. 19), The
equation (XIII.33) then gives

,,„ = ^^^' and r. = („ ^ i);p„ ^ 1) PfhU. (Xin.35)
For an indifferent mass structure the equation (XIII. 19) gives

for the velocity distribution above a rough surface. In Co=(^/7-35) the quantity
kisa measure of the roughness height of the bottom. Since for z=hi, u must be equal
to U the ratio TJp can be expressed in terms of U

• ^^ = {5-75 log (/;i/co)F (XIII.37)

and from (XIII.36)

10g(z/Co) /YTTT'58^

" = ^ 1 7rT~\ • (XIII.38)

log (hjco)

This gives a second equation for u and both must give a curve of the same shape.
The most suitable assumption is that both give the same values for the transport

General Theory of Ocean Currents in a Homogeneous Sea

411

(current amount) which will lead to the same value for (U — u) dz. From this a

relationship between h^ and n is obtained having the form

h
log — = («+l)loge.

(XIII.39)

Putting the expressions for T^ equal in (XIII.35 and 37) gives a further relationship
between h^ and U

A, = 0-160 4?^'^. (XIII.40)

n{n + 1) /

This relation shows that h-^ is directly proportional to U as was to be expected. With
this all the unknowns are determined.

Numerical values can be obtained in the following way : for a given value of n,
which according to equation (XIII.35) fixes the angle a, and for given latitudes 4> and
velocities U, the equation (XIII.40) allows to compute the related h^ and (XIII.39)
gives the value for Cq. From Cq the roughness height k can be found quite simply and
finally (XIII. 37) gives the value of T^. This then fixes the current structure completely.

Table 118 presents corresponding values for different roughness values of the sea
bottom as they could be expected to occur in reahty. These values are valid for ^=50°

Table 118. Basic values for the structure of the bottom current
{according to the Prandtl theory); 0=50°, C/=100 cm/sec

33i=

290

26°

Angle of deflection

56i°

61°

64°

Frictional depth h ""j

Cq >in metres
Roughness height k J

174
0-43
3-1

94
0031
0-23

76
00034
0125

Friction velocity u* (era/sec)

6-7

5 0

(/= 1-016 X 10~^ sec~^) and for [/= 100 cm/sec. The three roughness values correspond
to average conditions. The frictional depths are obtained in a row as 174, 94 and 76 m
which are plausible values. The vertical velocity distribution of u and v is shown in
Fig. 176 for the first case (« = 5, /?=174m) together with a vectorial representation
(uv); for comparison with the values given by the Ekman theory the corresponding
curves are shown by the dotted lines. The greatest differences, as would be expected,
appear in the immediate vicinity of the bottom; up to about 5 m from the bottom the
velocity increases linearly with distance from the bottom as was assumed by Ekman
to be the case inside his boundary layer.

Similar considerations also apply for the drift current caused by the wind. Here U
must be zero in equation (XIII. 37) and in addition the boundary condition at the
surface (~=h) must be

412

General Theory of Ocean Currents in a Homogeneous Sea

-* « • -t ai

= /^i8.

Thereby jS was presumed to be the direction of the wind stress. At the same time it
should be noted that under the influence of the turbulence of the wind the frictional
coefficient is largest at the sea surface and decreases with depth. In order to satisfy

280

240

200

' 1

160

120

'~'-\

'<<

1
1

■X

.---

.-'-

,-'■

02 0-4 0-6 0-8

0-2 0-4

^^2*^

<.°B

0-2

■'^

100'

i'^

;°

l6oV

rJ-

02 04 Q6 08

Fig. 176. Turbulent bottom current according to Prandtl (full lines: n = 5, /?i = 174 m;
dotted lines: bottom current according to Ekman (see Fig. 172).

these conditions the vertical current structure in the drift current will diff'er from that
in the bottom current where the frictional coefficient converges to zero at the bottom ;
it will have a similar form as compared with that shown in Fig. 172.

A theory of drift and gradient currents based on similar principles was put forward
by RossBY (1932) and later extended by Rossby and Montgomery (1935). This was
based on the principles of the newer turbulent flow theories and introduces in place
of the earlier used frictional coefficient the Prandtl mixing length. In drift currents this
is largest in the surface layers where the intensity of movement is greatest and decreases
with depth to vanish at the frictional depth. The theoretical treatment of this assump-
tion is very complicated and the results can only be shown by means of tables. Also
here the deflection angle of the wind drift comes out to be dependent on both the wind
speed and latitude, while according to the Ekman theory it should have a constant
value of 45°. The ratio of the velocity of the surface current to the wind speed (wind
factor, p. 418) results as equally dependent in a rather complicated way on the same

General Theory of Ocean Currents in a Homogeneous Sea

413

quantities. Table 119 gives some values for these relationships. A comparison of these
results with those of the Ekman theory and with observational data is given later on
(see p. 418). The introduction of a mixing length decreasing with depth and vanishing
at the frictional depth should give a correct representation of actual conditions only if
the turbulence arises solely from the wind drift and not from other currents which may
be present (for instance, tidal currents, gradient currents). If such influences exist, it is
necessary to introduce in the theory of wind-drift currents the vertical distribution of
the turbulent coefficients which corresponds to the total current. This, however,
modifies in turn the results. At present the Ekman theory appears to be a perfectly
satisfactory approximation to actual conditions, as long as our knowledge about the
vertical distribution of turbulence is not increased.

Table 119. Deflection angle and wind factor as a function of latitude and wind speed
according to the theory of Rossby and Montgomery (1935)

Defl

ection an

gle in deg

rees

Wind fac

tor VqIw

Wind speed w m/sec.

<!>: 15°
30°
45°
60°

350
38-6
40-6
42-0

38-7
42-8
45-4
46-8

41-1
45-7
48-4
50-2

43 0
480
50-9
52-7

00317
00292
00280
00273

00291
0-0268
00256
0-0249

00276
00254
0-0243
0-0237

00266
00245
00234
00228

(d) The '' Element ar'^ Current

In a homogeneous ocean no currents are possible other than drift and gradient
currents; at every point the steady current is made up of a pure wind drift and a pure
gradient current. These can be superimposed without mutual interference since each
component is entirely independent of the other. If the depth of the sea d is larger than
the upper and lower frictional depths D' and D", the resulting current system can be
separated into three current layers (see Fig. 177, left-hand side).

(1) The bottom current from the sea bottom to a height D" (lower frictional depth).

(2) The deep current from the level D' (the upper frictional depth) to the level D"
(the lower frictional depth).

(3) The surface current which is the resultant of the uniform deep current and the
pure drift current generated by the wind.

'I Su

rface current

Deep current

D"{~ B(

TTTTTTTTTZ

Bottom current

Wind

Fig. 177. Vertical structure of the "elementar" current (according to Ekman).

414 General Theory of Ocean Currents in a Homogeneous Sea

This vertical current stratification was termed by Ekman the ''elementar" current.
In limited seas the condition of continuity must also be satisfied. For stationary
conditions where everything remains invariable with time the inflow and outflow must
balance for a given oceanic space. The drift current is determined by the wind, thus
the slope of the sea surface and hence the gradient current must be such as to maintain
the constancy of the current system in time. The continuity equation and the boundary
conditions in this way determine the structure of the "elementar" current. A simple
case can be taken to illustrate these conditions (Fig. 177, right-hand side). A wind
parallel to a long straight coast will produce a drift current through which a total
water transport away from the coast down to the upper frictional depth is initiated.
This causes the surface of the sea to lower along the entire coast and will thus produce
a gradient current. The uniform deep current extending downwards from the surface
to the lower frictional depth D" will run parallel to the coast and thus cannot com-
pensate the removal of water away from the coast accomplished by the wind current.
This compensation must be provided for by the bottom current which carries water
towards the coast in the direction of the pressure gradient. The slope of the sea surface
will thus increase continuously, until the removal of water from the coast, due to the
drift current, is exactly balanced by the bottom current. The current in the top layer
will then be a vector composition of drift and deep current. The angle of deflection
at the surface will thus decrease from 45° to 18°. The current vectors are shown in
Fig. 177 for depth intervals of 0-2Z), with the same for the bottom current (at D' = D").
The uniform deep current occupying the deepest water layer between surface current
and bottom current is shown by the thick arrow; it is non-divergent and because of its
thickness is the decisive current component for the water transport in the oceans.
Further interesting cases of "elementar" currents in oceanic regions of special shapes
will be discussed in the following section.

It is of some interest to deal in some detail with the diagrams of forces for the three
layers of "elementar" currents. Since the vectors of Coriolis force and gradient force
are fixed by the current vector at the point under consideration, and by the sea surface
slope the primary task is to fix the frictional vector. This can be done in the following
way. If the current vector is denoted by t) (components u and v), the vector of the
deep current by 33 {U,V) and the difference vector by lu (h'^., n'j,)=(tu— 3.^).
(m— U, V— V), then the equations of motion will have the form

~f^= - Sq^-^ J^x and fu=-g~-i-Ry,

whereby -i^(/?a;, Ry) is the frictional vector.
However, for the uniform deep current

-fV=-g~ and fU=-g^.
■' dx dy

Subtraction gives

— fWy = R^ and fw^^Ry

so that w'x Rx + »*'i/ Ry = 0.

This, however, is the necessary condition for the vector of the frictional force
^{Rx, Ry) to be at right angles to the direction of the difference vector tu. Thus the
direction of the vectors of all three forces involved are known and therefore a diagram

General Theory of Ocean Currents in a Homogeneous Sea

415

offerees for each layer of the "elementar" current can be constructed. Figure 178 shows
these force diagrams for always one level of the three current layers. In the surface
current the frictional vector is directed to the side of the gradient vector pointing in the
direction of the water movement, and rotates in a clockwise direction with decreasing
intensity when going downwards and vanishes at the frictional depth. In the deep

(b)

(c)

Fig. 178. Schematic diagram offerees for three levels of the "elementar" current (Northern
Hemisphere): {a) surface current, (Jb) deep current, (c) bottom current. OG, OC and
OF vectors of pressure gradient, of Coriolis force and of frictional force; v = velocity
vector in the level under consideration; V = velocity vector of the deep current;
w = vector of the velocity difference: v — V.

current, gradient and Coriolis force balance each other without any frictional effect.
In the bottom current the frictional vector is directed to the side of the Coriolis force
pointing more or less in the opposite direction to that of the velocity and rotates
anticlockwise while approaching the bottom. From this distribution it can be realized
that in the surface current the frictional vector corresponds to a driving shear stress
which takes its strength at the sea surface from the energy of the wind, while in the
bottom current it indicates the retarding effect of the underlaying bottom topography
(break on the motion).

(e) Drift and Gradient Currents according to Observations; Piling up of Water by
Wind {''Windstau'')
The two parts of the "elementar" current are never developed in the ocean in pure
form and it is to be expected that pure drift currents in the ocean will always be some-
what masked by the effects of superimposed gradient currents. It will therefore not be

416

General Theory of Ocean Currents in a Homogeneous Sea

easy to test the properties required by the theory. Three consequences of the theory
are possibly most suitable for such a test:

(1) the deflection of about 45° cum sole from the direction of the wind which is
almost independent of latitude (except near to the equator) ;

(2) the restriction of penetration of the drift current by the frictional depth D;

(3) the dependence of the sea surface velocity of the drift current on the shearing
stress of the wind.

Angle of deflection. By special selection of oceanic areas, where it would be expected
that the wind alone would be decisive in determining the currents, Galle (1910)
showed that the deflection required by theory was actually present. For this he used
the large amount of data available for the Indian Ocean for all November months from
1858 to 1904 between 20° N. and 50° S. and 10° E. and 130° E. Taking together two
degree zones in each ten-degree field, the theoretical deflection to the right was obtained
in 77% of all cases in the Northern Hemisphere and in 69% in the Southern Hemi-
sphere. Three areas were examined with particular care: the sea between Socotra and
the Maldives, the South Equatorial Current and the west wind drift of higher southern
latitudes. Table 120 shows average values for larger areas. The mean of all values is
about 46° and in fact there seems to be no dependence on the latitude; both these
circumstances are in accordance with the theory for a constant frictional viscosity
coefficient. Forch (1909) used the survey on wind and current conditions in the Eastern
Mediterranean published by the "Deutsche Seewarte" to obtain an estimate of the

Table 120. Mean angle of deflection in the Indian Ocean (cum sole)

in all cases

5°-20° N. 50-60° E.
60^-70° E.

62°

44°

40°-50°S. -

10°-20° E.
20°-30° E.
30°^0° E.
70°-80° E.
80°-90° E.

55°
41°
42°
41°
43°

10°-20°S. 70°-80°E.
80°-90° E.

47°
51°

Table 121. Mean angle of deflection in the Eastern Mediterranean (cum sole) in all

cases

Area

36=-38°N.
15°-20°E.

34°-36° N.
15°-20°E.

34°-36° N.
20°-25° E.

32°-34° N.
25°-30°E.

Annual mean

38-2°

■ 33-1°

52-4°

430°

Mean for the
four fields

Jan. /Feb. Mar./Ap
AAV 45°

r. May June/July Aug./Sept. Oct. /Nov.
86° 47° 23° 23°

Dec.

45°

Mean
411°

deflection of the current from the wind direction. The differences between wind and
current azimuth for the four larger areas are given in Table 121 as annual average
values derived from the monthly means. The mean of these rather scattered values is
around 42° cum sole. In the annual variation the angle is nearly 45° from December
to April, reaches a very high value in May and then during the warmer part of the
year from August to November is about 20°. It is possible that the strong surface

General Theory of Ocean Currents in a Homogeneous Sea

417

density gradient during the summer gives rise to a strong differentiation in the magni-
tude of the frictional coefficients in a vertical direction v^hereby the angle of deflection
is reduced.

Even more penetrating investigations have been made of the deflection angle in
shallow seas (lightship observations). These values have, however, mostly been made
in coastal areas or over large banks where disturbances can be expected but these
can be eliminated by special grouping of the data. According to the Ekman theory
there will be no strong deep currents in any largely enclosed sea (see p. 428). A com-
parison between theory and observation can then be made in such a case. For a shallow
sea (depth d) the theory requires the deflection to be smaller the smaller the ratio d.D.
On the other hand, the thickness D of the drift current will increase with increasing
wind strength. It can thus be expected that in a shallow e?iclosed sea, the angle of deflec-
tion will become smaller as the wind increases. From data on currents recorded by
Finnish light-ships, Witting (1909) found that the angle of deflection was always
cum sole and that it could be expressed by the relation

a = 34° - 7-5 Vw,
where u' is the strength of the wind in m/sec. The strong ellipticity of the current ellipses
at the different lightships indicates a preferred current direction caused along the
longer axis of the sea which certainly affects the results. Qualitatively, however, it
corresponds fully to the requirements of the theory. Also Dinklage (1888) obtained
similar results from observations made at the Adlergrund light-ship (Baltic).

The question of testing the Ekman theory has been discussed in detail by Palmen
(1930 b, 1931) in connection with an evaluation of the currents in the northern part
of the Baltic. This was based principally on observations made at the rather openly
situated Swedish lightship "Finngrundet" (60-0° N. 18-5° E. at the southern end of the
Gulf of Bothnia) for the period 1923-27. Tables 122 and 123 show clearly the relation-
ship between wind and current on the one hand for different wind strengths and on
the other hand for different wind directions. These correspond rather well to the
requirements of the theory. Especially the confirmation of the turn of the current
direction with increasing depth deserves our attention because only few observations
of that kind are available. After elimination of non-significant disturbances the
following corrected values are obtained for wind strengths of 4-5 Beaufort:
Vo = 9-2 cm/sec, ao = 35°, KgoiKo = 0-76;

K,o = 7-0 cm/sec, ajo = 54°, Aa = 19°.

Table 122. Currents at different wind strengths at the lightship ''Finngrundel'' (Gulf of
Bothnia, 1923-27) (according to Palmen)

Wind strength (Beaufort)

2-9

3-9

4.9

5-9

6-8

7-8

9 0

9.9

Vq (cm sec)

5-8

8-4

11-3

12-3

14-7

19-2

22-9

27-3

F^o (cm sec)

1-6

2-2

4-5

6-2

9-6

10-2

130

18-3

19-7

24-1

V,o:Vo

0-87

0-71

0-78

0-74

0-85

0-83

0-88

0-95

0-86

0-88

Deflection a^

26°

41°

38°

33°

34°

35°

32°

25°

36°

8°

O-20

32°

50°

48°

42°

41°

45°

52°

38°

40°

11°

"20 - ao ■

6°

10°

7°

10°

20°

13°

4°

3°

418

General Theory of Ocean Currents in a Homogeneous Sea

Table 123. Currents for different wind directions at the lightship ''Finngrundet" (mean

value at 2-7 Beaufort)

Wind direction

N.E.

S.E.

S.W.

N.W.

Mean

Vq (cm /sec) ....

8-6

11-2

121

8-7

7-5

9-2

7-4

8-2

9-2

K20 (cm /sec) ....

6-8

9-6

11-5

6-8

5-5

7-9

5-7

7-6

K20 -yo-

0-79

0-86

0-95

0-70

0-73

0-86

0-77

0-85

0-81

«o ....

30°

35°

41°

40°

38°

22°

34°

35°

020 ... .

39°

46°

47°

46°

55°

50°

41°

47^

46°

020 — Oo

11°

6°

5°

15°

12°

19°

13°

11°

The directional turn between 0 and 20 m depth is 19° cum sole and at the same time
the velocity falls by about a quarter of the surface value. This turn of the current
is in good agreement with the theory; the decrease in velocity is, however, much too
small to be explained by a constant frictlonal coefficient ; for a water depth of 23 m
and for a 77 about 200-300, it must be about 0-12 instead of 0-81. Only an assumption
of a variable r] with depth approximately in the sense of the discussion given on p. 405
could explain such a small decrease.

The relationship of wind strength to current strength. According to the theory the
surface velocity Vq is given by the relation

Ko = ..^ ^ . ,. . (XIII.4])

From this it follows that for constant 77 and p the surface velocity Vq is proportional
to the wind velocity w and is inversely proportional to the square root of sin ^:

Vq = —-^ w (XIII.42)

V(sm (p)

A is a universal constant. The quantity VqJw is denoted as the "wind factor". Numerous
investigations have been made of this relationship (see especially Thorade, 1914);
the following values have been found for A, when Vq and h' are expressed in cm/sec:

Mohn
00103

Dinklage
C-0127

Witting
00100

Thorade
00126

Pal men
00114

Nansen
00190

Sverdrup Brennecke
00177 00269

The first of these values are in good agreement. For the ice drift, on the other hand,
considerably higher values were obtained (Nansen, 1902; Sverdrup, 1928;
Brennecke, 1921). See p. 437 concerning these. Usually an almost linear relationship
has been found between the wind velocity and the velocity of the surface current.
Witting and Thorade, however, arrived at a different result : for a wind force of up
to 3 Beaufort a better fit to the observations was obtained by a quadratic relation.
Palmen believed, however, that this was due to the uncertainty of the conversion of
wind strength from the Beaufort scale into m/sec. For the magnitude of 77 it seems to
be also of importance, on what height the wind measurements are based; a better
agreement could probably be obtained if also this was taken into account (Exner,
1912; Durst, 1924).

General Theory of Ocean Currents in a Homogeneous Sea 419

The shearing stress of the wind and piling up of water caused by the wind. There are
two ways in which the wind stress can be determined. The first is afforded by equation
(XIII.41). This requires a knowledge of the frictional coefficient iq, but its dependence
on the wind strength is not well-enough known. Ekman has indicated a second possi-
bility using the piling up of the water ("Wasserstau") by the wind and using the current
produced by the wind over a confined sea. If the effect of the Earth's rotation is dis-
regarded (/= 0), and if dpjdx is replaced by the slope / of the sea surface, then the
first of the equations (XIII.28) for a variable -q gives the equation

d I cti]

This can be integrated considering the boundary conditions

= -T and (m),=<j = 0
and taking into account the continuity equation

('9

2 = 0

u dz = 0.

The frictional coefficient t] increases strongly with distance from the sea bottom.
Using the relationship introduced by Fjelstad (see p. 405)

'» (' - ?T-J

where « is a positive number smaller than 1 and e is a very small and positive number
as compared with d, then the integration, neglecting small terms, gives an approxi-
mately valid relation (Palmen, 1932, 1933)

3 — « T
1=--^—.. (XIII.43)

2 gpd

For a constant frictional coefficient (« = 0) it transforms to

i=-l ~. (XIII.44)

2 gpd

This equation applies for stationary conditions and a constant density. In the ocean
the water is stratified and the wind itself gives rise to changes in the oceanic structure.
Thereby solenoid fields are generated and the use of the formulae under these real
conditions must necessarily lead to difficulties. To avoid these, Ekman and Palmen
(1936) therefore reformulated the equation (XIII.44)

i = - -, , (XIII.45)

gpa

where e is always smaller than 3/2. Assuming that there is no bottom friction (gliding),
then e = 1 ; when the depth is large (greater than D) this is only approximately true.
If there is adhering ("Haften") of the water at the bottom, then e = 3/2. It is not
possible to determine € in each case ; if e = 1 , then T is somewhat too large at shallow

420

General Theory of Ocean Currents in a Homogeneous Sea

depths. Since, however, due to the dependence of the frictional coefficient -q on the
depth, the stress T is somewhat too small it is of no great importance if e is put equal
to 1, especially for more intense winds.

Most important, therefore, is the determination of /. This slope is made up of three
components: the first depends on the direct piling up of water by the wind, the second
is the static effect of the atmospheric pressure distribution, and the third is due to the
deep current produced in the enclosed basins by the wind (current effect). The atmos-
pheric pressure effect can be eliminated quite simply (pt. I, p. 7) ; the current effect
depends in the first place on the boundaries of the basin and on the stratification of
the water in it. In elongated seas with strong stratification (such as the Gulf of Finland)
it is rather large and acts at right angles to the main direction of the current. In an
oceanic area without any particular major axis the greatest piling up occurs exactly
in the direction of the wind (for example, in the Gulf of Bothnia).

The equations (XIII.43-45) were first applied by Ekman (1905) for the case of a
storm in the southern Baltic (Colding, 1881) and gave 7=3-2 X lO^^v^ {w in
cm/sec). Inserting the density of the air, p' = 1-25 x 10-^ gives

r=2-6 X 10-=^p'm'2.

This relation applies for wind speeds of up to 20 m/sec. The magnitude of piling
up by the wind is given in Table 124. In more recent investigations Palmen has deter-
mined the dependence of the piling up by the wind on the strength of the wind and the
depth of the water for the Gulf of Bothnia from observations of the water level. He
found that, for the water depths in the area under investigation, the "Windstau" was
directly proportional to the wind intensity for lighter winds, while for strong winds
was rather proportional to the second power of the wind strength. Furthermore, the
tangential pressure of the wind according to equation (XIII.45) could usually be
expressed by the formula

r= 0-14 X lO-V + 0-022 X 10-''vv'2.

Table. 124. Piling up of water ''Wasserstau' by the wind for a depth of 50 m

(according to Palmen)

Wind in m/sec

15 20

Filing up of water (cm /1 00 km) .

007 0-59

1-65

6-6

14-9 26-4

41-3

59-4

In a later investigation Palmen and Laurila (1938) found

id =3-15 X 10-V2

for rather intense winds during a storm in October 1936, which leads for a mean water
depth of 50 m and p' =--- 1-3 x 10-=^ to

r= 2-4 x 10-3p'vf2.

The values for the constant k agree well with this (see equation X.9). A more recent
determination in a similar way was made by Hela (1948), who found ^ = 1-9 X 10-^
[^cm""^ sec"2].

General Theory of Ocean Currents in a Homogeneous Sea 421

According to recent hydrodynamic theory (see for instance, Prandtl, 1942,
p. 108) the investigations of flow over smooth and rough surfaces have shown that
the shearing stress of the wind follows the relations:

w zp

for a smooth surface: ,, , ,, = 5-5 + 5-75 log — ^J{r\p') (XIII.46)

and

z -\- Zq
for a rough surface: w = 5-75 Vi'^lp') log — :: — • (XIII.47)

To decide whether a water surface is considered "smooth" or "rough" for different
wind conditions it is necessary to investigate the vertical wind distribution over it.
This has been done by WiJST (1920) and by Rossby and Montgomery (1935), who
have discussed the results and have concluded that for winds of more than
6-8 m/sec (measured 1 5 m above the surface, Beaufort 4) the water surface must be
considered as "rough". As a result it was ascertained that for moderate and strong
winds the roughness length z^ was independent of the wind strength and had a constant
value of 0-6 cm. The formula (XIII.47) then gives

r=2-9 X \0-^ p'n'l, (XIII.48)

where n\o is the wind speed at 10 m above the surface. This formula, however, no
longer applies when vvjo < Beaufort 4 or 6-8 m/sec and the surface has to be con-
sidered as "smooth". In this case the formula (XIII.46) will be valid. The values of T
calculated in this way are about a third less than those computed from (XIII.48).
As a reasonable first approximation they satisfy the relation

r = 0-9 X 10-3 p'wlo. (XIII.49)

This shows that there is a laminar boundary layer of small vertical extent in wind
profiles above the water surface, which reduces friction considerably (Rossby, 1936 b).
Further analyses of measurements of the tangential wind stress and the rouglmess of
the sea surface have been made by Neumann (1948) who showed that the frictional
factor at the surface decreases with increasing wind speed and that in general at the
surface of the sea

r=0-9 X 10-3p'i;3/2.

Neumann attempted to explain this striking behaviour of the hydrodynamic roughness
at the sea surface by changes in the nature of the sea-way dependent on the wind
strength. The waves move with the wind and the surface of the sea will very likely
tend towards a profile, offering the least possible resistance to the wind over it
(Model, 1942); see Munk (1955) for a more detailed discussion. Further measure-
ments of the wind stress over water have been made by van Dorn (1952).

Another method for the determination of the wind stress on the water given by
Shepard and Omi (1952) making use of the geostrophic deflection of the wind at the
sea surface and upwards to a height of some hundred metres above it. The geostrophic
wind can be calculated with sufficient accuracy and the deviation of the observed
wind from this depends only on the friction. This method gives resistance coefficients
about 1 X 10-3.

422 General Theory of Ocean Currents in a Homogeneous Sea

A summary of all values of the resistance coefficient shows that the stress can be
represented by a formula of the form

where n may differ somewhat from 2 or /c itself is a function of u-. For wind speeds of
up to 10 m/sec the values of k are very scattered and it is not easy to decide whether
this scattering is due to errors in measurement or due to effects which have not been
taken into account (such as the vertical stability of the air mass over the water or
deviations from the steady state or stratification of the water and others). The dis-
continuity imagined by Munk at 6-8 m/sec has not yet been confirmed and no definitive
relationship between the stress and the wind can be obtained at the present time,

Frictional depth and frictional coefficient. According to equations (XIII.26) the
frictional depth depends on the wind stress T and on the surface velocity V^. For
T a dependence of the form (XIII.48) can be taken with an average coefficient of
2-9 X 10-3 p' ^ 3.5 X io-« ; Fo is related to the wind speed w by (XIII.42) (A approx.
0-0114). Tand Vq can be eliminated in this way from the formula

7tT

\/2' Vq pco sinrf)

giving

7-6hr
D = ,, ■ ,, , (XIir,50)

V(sm <^) '

where \v is given in m/sec and D in m. If Vq is retained, a very simple formula results
which was already derived by Ekman

D = 670 Fo (XIII.51)

which is very useful for the estimation of D. This states that the frictional depth is
approximately equal to the distance travelled by the surface water in a pure drift
current in about 600 sec or 10 min. It should be noted that equation (XIII.51) does not
involve the latitude. Thorade (1914) derived the equation

\/(sm 4>)
for wind speeds less than Beaufort 3 (about 6 m/sec). All these formulae are of course
only approximations, since at the present time systematic current measurements
from which accurate values could be derived are not available.

Observations on the thickness of drift currents are usually in general agreement
concerning magnitude with the values given by formula (XIII. 50). The oceanic struc-
ture in the region of the North and South Equatorial Currents in the Atlantic Ocean
indicates that the wind current here has a depth of about 150 to, at the most, 200 m and
and thus that the frictional depth in these latitudes only barely reaches these values.
Towards higher latitudes it decreases. Brennecke (1921) found a frictional depth of
about 50 m during the ice drift of the "Deutschland" in the Weddell Sea and Sverdrup
(1928) has shown from Brcnnecke's values that there is an increase with increasing
wind speed as is shown by the following values :

Drift velocity (cm/sec) : 5.52 9.81 14.85 24.60

Frictional depth £> (m) : 45.6 56.2 (39.1) 69.1.

General Theory of Ocean Currents in a Homogeneous Sea 423

Using the equations previously derived to calculate 77 gives

77 = 1 -03 vv^ for IV < 6 m/sec,

and 77 = 4-3 u'^ for vv > 6 m/sec.

The values calculated from these formulae are also to be regarded as only approximate
average values; the few directly determined values are widely scattering and indicate a
large dependence on the vertical stratification of the water masses, Schmidt (1917)
has presented some values :

Wind speed (m sec)

■q (cni-^^ g sec"^)

(1)

110

220

430

1720.

The high values for strong winds apply of course only for the especially intense tur-
bulence produced by the wind in the uppermost water layer; below this layer the co-
efficient decreases rapidly with depth. An average value for the top layer of the ocean
will be between 50 and 100. Its magnitude in the deep layers will be about 1-10,

Diagrams of forces for a wind-driven, stratified ocean. With a complete knowledge
of the total current and pressure structure of the ocean diagrams of forces for any
layer can be derived in the following way (Defant, 1941 b). Denoting the sea surface
slopes (of the isobaric surfaces in the deeper layers) in the positive .v-direction (towards
east) with i^ and in the j-direction (towards north) with iy, then the equations of
motion for a variable 17 are of the form

8 / 8u\ 8 / 8v\

fpv + gpi. + ^, [1 -^.j = 0; -fpu + gpiy+ ^ [r^ j^j = 0. (XIII.52)

Integrating these equations from the surface to the depth D with the assumption that
the current falls to zero at a depth d and taking furthermore into account that for
z == 0 the components of the wind stress are given by

cu 8v

and vanish when z = d, the following equations are obtained :

f7v + g'pi'x+T, = 0 and -f^u -{- gJTy + Ty = 0, (XIII.53)

where the integrals (sums) down to the depth d are indicated by a bar. This states
merely that for a steady current the Coriolis force must be in equilibrium with the
sum of the total pressure force and the total wind stress exerted on the entire layer.

The equations (XIII.53) can be evaluated numerically from the absolute topography
of the pressure surfaces and of the physical sea-level, as well as from the rather reliable
vertical current distribution as measured at two anchor stations in the region of the
South Equatorial Current in the Atlantic. Table 1 25 contains all the necessary numerical
values and Fig. 179 shows the vertical changes in current- and pressure-gradient
quantities for calculation of the integrals. It can be seen that the £'-component of
the velocity decreases regularly with depth, while the A^-component changes already
in the uppermost layers from small positive values to negative values and then falls
back to zero at 100 m. This distribution leads to a turn of the current vector cum sole
which must be the case in drift currents. Below this there is only a gradient current

424

General Theory of Ocean Currents in a Homogeneous Sea

100
200
300

400
500

/? X dyn cm
0-2 0-4 06 08

10 12

"~~^

^pA

\ y

Ipv

-30 -24 -18 -12 -6
p, cm/sec

Fig. 179. Vertical changes in the pressure gradients and of the velocity components in the
central part of the South Equatorial Current in the Atlantic Ocean.

which, however, also disappears at 500 m depth since there the isobaric surfaces become
almost horizontal.

Table 126 gives integral values for the equations (X1II.53) and the corresponding
resultant values of the wind stress; Fig. 180 presents the diagram offerees for this cen-
tral part of the South Equatorial Current. The average direction of the south-east trade
wind during the observational period was S. 40° to 45° E. and the mean wind force

Fig. 180. Schematic diagram of the forces in the South Equatorial Current in the South

Atlantic Ocean.

about 12 m/sec. This wind direction is in excellent agreement with the direction of the
wind stress. The wind stress can be calculated from the meteorological data using
equation (XIII.48) or from the oceanographic data using equation (XIII. 37). In the first
case wind stress and wind speed lead to a constant value for A' of 2-5 x 10~^ which is

General Theory of Ocean Currents in a Homogeneous Sea

425

Table 125. South Equatorial Current in the Atlantic Ocean
(approx. 14° S., 20° W. to 8° S., 15° W.)

Pressure gradient

dyn cm,

100 km

Vertical current distribution

p (dbars)
depth (m)

.d/7 100 km

U V

in situ

Direction (dyn. cm)

P'x

P'v

cm/sec

N. 60° E.

0-97

0-86

0-49

-32

+ 7

-32-9

+7-2

24-3

—

-14

-8

-14-3

-8-2

24-7

100

N. 30°E. 100

0-51

0-89

+ 9

+ 1

-9-2

+ 10

25-9

200

N. 20'E. 1-20

0-42

116

-4

-1

-41

-10

27-7

500

0 000

000

(+1)

+ 1-0

29 4

Table 126. Diagram of forces in the South Equatorial

Current of the Atlantic Ocean

(Forces in dyn/cm^)

Coriolis force

Pressure force

Wind stress

-/or = +1-77

+fpli= -6-73

SI5°E 6-95

gPl^= +1-49
gpt\ = +3-28
N24°E 3-51

r„ = -3-26

Ty = +3-45

N 43° W 4-74

in good agreement with the known values. Alternatively, taking h-^ (the frictional depth
of the drift current) as about 200 m, the roughness parameter Cq as 0-3 and the surface
velocity U as 35 cm/sec, equation (XIII. 37) gives exactly the required value of 4-74.
These calculations show in any case that the oceanic current conditions are in good
agreement with hydrodynamic concepts about the driving forces.

The dissipation of the current energy in the ocean. It is probably of some interest to
calculate the amounts of energy dissipated in a drift current due to the apparent
friction. The energy consumption is of course largest in the uppermost layer and
decreases rapidly with increasing depth. If only the /o/a/ energy consumption is required
this can be calculated rapidly in the following way. The total work done in the interior
of the water must be supplied from the wind at the sea surface. This is, however,
given by force x distance. The force is the wind-stress component in the direction of
the surface current; the component at right angles does not enter into the calculation.
This component is Tcos 45 "" and the distance travelled in unit time is Vq. The energy
consumption per second in a vertical water column of 1 cm^ cross-section can then be
obtained using equation (XIII.26) (Schmidt, 1919) and is given by

W = Vq \/{t]pw sin (/«).

The values of tj given by Thorade give the energy values shown in Table 127. In a
vertical water column the total work expended should lie between 2 and 40 erg/sec.
There is a considerable increase in these amounts with increasing wind speed and the
latitude also has an appreciable effect.

426

General Theory of Ocean Currents in a Homogeneous Sea

Table 127. Energy Dissipation in Ocean Currents

(according to Schmidt)
(Values in erg cm"- sec~^)

Wind speed w (cm/sec)

10°

4-5

230

550

Latitude

40°

2-3

1-1

120

290

70°

1-9

6-3

100

240

(/) The Effects of Coasts on the ''Elementar''' Current

The vertical structure of the "elementar" current depends essentially on the direction
of the wind relative to the general outline of the coast, since this has a large effect on
the equation expressing the condition that for stationary conditions the transport
component at right angles to the coast must be zero. Ekman (1923) has presented a
solution in two simple and very instructive cases. The first case assumes an extended
oceanic region off a long straight coast over which blows a wind of constant force
and direction. The water depth d is assumed to be constant and greater than 2D. The
sea-level will fall uniformly from the coast towards the open sea and the pressure
gradient produced by the piling up of water by the wind ("Windstau") will be at right
angles to the coast. With an arbitrary orientation of the co-ordinate system the trans-
port components M'^ and Afy will be given by equation (XIII.27). The transport
components of the gradient current are given by

M'^ = bU^ - BUy and W; = BU^ + bUy, (XIII.54)

whereby U^ and Uy are the components of the uniform deep current and

(- - S)

5 = V- and b = \ pd

If the X-axis is oriented along the coast, then f/,, = 0 and from continuity equation
M'y X M"y = 0 is obtained

Bf pcoD sin 0

r..

For a given T and a given angle between wind and coast the drift current and the
gradient current is fully determined. Ekman has given a simple graphical method for
the construction of the total current structure in this case. Figure 1 8 1 shows this current
structure in some special cases. The current arrows have to be visualized as drawn
from the point o to the points on the curve and the small points refer to heights of
0-1, 0-2 D etc., above the sea bottom and to depths of 0, 0-1 D, 0-2 D etc., below the
sea surface. The wind direction is indicated by the arrow. The cases correspond to
angles of /S = 0, +45° and -45°.*

There is a considerable difference between conditions when the water flow is un-
hindered in all directions or when it is adhered due to any kind of influence. In the

* ^ = 0 indicates a wind direction parallel to the coast ; the increase in j3 is positive to the right and
negative to the left.

General Theory of Ocean Currents in a Homogeneous Sea

All

Fig. 181. Vertical structure of the "elementar" current for different orientations of the coast
relative to the wind (according to Ekman) (the arrow indicates the wind direction).

first case only a pure drift current is formed and the effect of the wind is restricted to
a relatively thin top layer. At coasts, however, the effect of the current-producing
wind extends almost down to the sea bottom due to the generation of deep currents.
Their velocity is not insignificant and may be as much as half of that of the surface
current. The second case is that of a sea enclosed by land, with a wind of constant
direction and constant speed blowing over its entire surface. Here the continuity
condition requires that the transport in all directions should be zero, that is, that the
total gradient current transport must be the same as that of the drift current and
directed oppositely. The boundary condition equations are now

^/■x + ^x = 0 and My + My = 0.

Taking the positive j'-axis along the direction of the wind stress, then Ta- = 0 and
Ty = T. This gives

Tlf-i-bU^-BUy^O and BU^ + bUy = 0

from which it follows that

bT . __ BT

U.= -

and Uy =

f(b^ + B') ^' f{b^ + B^)

If the angle {cum sole) between the gradient current transport and the pressure
gradient is denoted by fi and if Uy — 0, then

My^-B "°^ ^^tan-^.

This angle is almost 90°, if the depth of the sea is not too small (for djD = 1, 2, 10,
^ is approx. 79°, 85° and 89°, respectively). However

^ = -^=tana,

where a is the angle between the direction of the deep current and that of the wind, or
a — |7T is the angle between the directions of pressure gradient and wind. Since

428

General Theory of Ocean Currents in a Homegeneous Sea

a = TT — ^, this angle will be ^tt — /3 {cum sole). The velocity of the deep current is
then

T . ^ 27tT
sin p ^ -7^ cos p.

pfD

The gradient current now extends almost throughout the entire water mass, so that
even a low velocity of this current is sufficient to compensate the drift current trans-
port. The greater the depth of the water, therefore, the lower will be the velocity of
the gradient current, and the less will be the effect of the coasts on the surface current
given by the resultant drift and gradient current. As shown by the above equation,
containing cos ^ and the frictional depth D in the denominator, the deep current V
is very weak. Ekman has calculated numerically three special cases {d — 0-5 D,
d = \-25 and 2-5 D). Figure 182 shows the vertical current structure in the usual way

d=<y^D

d--V2W .

cf^2-50

Fig. 1 82. Vertical structure of the "elementar" current in a water basin with everywhere closed

(according to Ekman) (the arrow indicates the direction of the water "stau" (direction

in which the water is piled up by the wind)).

The uniform deep current can be realized at greater depths, however, it is very weak
and at still greater depths vanishes almost entirely. The water is piled up nearly in the
wind direction in all cases and is therefore only slightly affected by the Earth's rotation.
This may be the reason for the late recognition of the effect of the Earth's rotation on
ocean currents.

( s) Effect of Bottom Topography

The results so far presented of the theory of steady currents in a homogeneous
ocean, of which the most important one is the derivation of the "elementar" current,
permit a considerable insight into ocean currents produced by the wind in a homo-
geneous sea; however, they can only be applied to smaller oceanic areas over which the
effects of latitude variation, as well as that of local variations in depth and wind can
still be disregarded. The further development of the theory by Ekman (1923, 1928 a,
1932 and Thorade, 1933/)) was devoted primarily to the uniform deep current,
and an investigation was made to determine the kind of change which occurs in the
deep current when the water masses transported enter

(1) into areas with non-uniform winds,

(2) into areas with varying depth, and

(3) into widely differing latitude regions.

Thereby conditions become rather complicated, especially with the additional
assumption that the upper and lower frictional depth vary, not only from place to

General Theory of Ocean Currents in a Homogeneous Sea 429

place but also with the velocity of the deep current. In that way the theory becomes
very complete indeed, but then in most cases the results do not allow a clear insight.
It is therefore necessary to investigate the effect of each factor separately.

The condition for a constant sea-level is that the total transport M, which is made
up of M' and M", the transport for the drift and the gradient current should satisfy
the equation:

div M' + div M" = 0. (XIII.55)

To this must be added the boundary condition along the coast (vertical coast down to
the sea bottom at depth d)

M'n + Ml = 0, (XIII.56)

where the index n indicates the transport components at right angles to the coast.
Disregarding differences in latitude and in the two frictional depths, then the equa-
tions (XIII.55, 56 and 57) after some calculation give the differential equation

8^C , s^C , g [dd ec . 8d en i /er er

dx^ + 8y^ + B [dx cy + dy 8x) ~ gB \ 8x 8y ) (Xni.57)

The effect of the difference in depth can be investigated more closely using this equa-
tion in special cases. A simple case is shown in Fig. 183 which represents a vertical
section in the sea directed along the .v-axis and parallel to the coast. The sea bottom
slopes downwards in the direction of the coast by D over a distance /, so that the
gradient is 8dl8x = Djl. It is necessary to investigate whether a deep current parallel
to the coast is at all possible. If the wind is assumed to be constant over the area
{cTyj8x = 8Txl8y = 0), then since 81,1 ex = 0 and since for p = 1 , DjB = In,
(XIII. 57) gives the differential equation

8^C 277 8C

ey^ + T8y = ^ (XIII.58)

the solution of which is given by

— = /^e-<2-')^ and U = Uoe-<^-'^)\ (XIII. 59)

where /q is the slope of the sea surface and Uq is the velocity of the deep current at the
coast. The latter decreases rapidly with distance from the coast, so that at a distance
hi Uq has fallen to ^'23 Uq. The deep current is limited to a narrow strip off the coast,
the individual current filaments perform a shearing motion relative to each other and
and observer on the sea would notice a vortex motion contra solem. Figure 183 shows
the assumed wind direction off the coast. The thin dotted line shows the decrease in
velocity for a frictional depth proportional to the velocity of the deep current.

For constant D and for a locally constant wind it is also easy to investigate how the
deep current is transformed when flowing over a sea bottom shaped like corrugated
sheet-iron. The outline of the coast and the wind direction are assumed to be at right
angles to the ridges of the bottom waves. The depth of the sea is then a function only
of X and with —8CI8y = i^ = const, and if the sea depth d = d^ + 8 cos (2ttII)x,
one obtains from (XIII. 57)

8^ 2tt8 Itt

T- = ~ ^ /'o COS -J- X.

ox D ^ l

430

General Theory of Ocean Currents in a Homogeneous Sea

Coast

Fig. 183. Upper picture: vertical cross-section parallel to the coast through an ocean with
increasing depth. Lower picture: horizontal section through the field of the deep current
(full line and arrows are valid for a constant frictional depth ; dotted curve and arrows are
valid for a variable frictional depth. The arrow at r indicates the assumed direction of the

wind stress.

From this it follows easily that

g Itt g8 2tt

U^ = Jo and Uy = j- ^ cos j x

and the stream lines are given by the equation

81 . In

y = -^sm -j- X -}- const.

At a sufficient distance from the coast the current field shows sine waves (Fig. 184) the
amplitude of which depends on the absolute size of the bottom waves. The depth of
the sea plays no role here; thus the velocity in the direction of the coast is constant,
but the total velocity is smaller than in a sea with a constant depth. At the same time
the stream lines deviate more and more from a straight course and take on a curvature
cum sole as the current passes over decreasing depth and the reverse (contra solem,
increasing depth).

The effect of varying latitude is shown principally by the fact that the deep current is
no longer exactly divergence-free. However, this divergence only becomes important
in lower latitudes, and in middle and higher latitudes it is always very small. Since in
lower latitudes the direction of surface currents is predominantly zonal, this should
also apply to deep currents and also here the effect of div U then remains small.

If all three of the factors influencing the deep current (wind field, bottom topo-
graphy and the Earth curvature) are considered at the same time the treatment becomes

General Theory of Ocean Currents in a Homogeneous Sea

431

more difficult. Instead of determining curl U, Ekman in his older theory (1923)
investigated a quantity W, termed the "quasi-vortex". It is strictly not identical with
curl U but in most cases agrees with it in sign and magnitude. This quantity W is the
sum of thiee terms

(XIII.59 a)

The first term depends only on the wind and is directly proportional to the vorticity
of the wind {anemogenic vortex effect), Wa depends on the slope of the bottom topo-
graphy but not on the total depth {topographic vortex effect), W^ depends only on the
curvature of the Earth {planetary vortex effect). The two latter effects are the most
important ; their mode of action has been illustrated in the examples previously dis-
cussed. When a current flows across the isobaths of the sea bottom, even quite small

Fig. 184. Deep current influenced by a wave-form sea bottom profile. Lower picture:
vertical cross-section parallel to the coast. Upper picture: horizontal section through the

current field.

slopes can affect the deep current and usually give it quite a different appearance. On
the other hand, the curvature of the Earth so strongly resists forced meridional water
movements that in the lower latitudes almost only zonal currents are possible. For the
combined topographic and planetary vortex effect Ekman obtained the same results
as were derived earlier for frictionless gradient currents (see p. 386). This suggests that
the simplifications introduced for their calculation eliminate the frictional effect to
such a degree that only the part for frictionless currents remains.

In a new theory Ekman (1932) extended his investigations, in which he still deals
only with steady currents. But previously these currents were also subject to the
condition of no acceleration dujdt = dvjdt = 0, while for a steady current only the
condition cujdt — dvjdt = 0 is required. Accelerations are thus possible due to the
circumstance that water elements are subjected to velocity change when changing their
position. These accelerations give rise to changes in the form of the current which may
be quite large. For example, the case discussed previously of a current over a wave-
shaped sea bottom (Fig. 184) would show two types of change: First, the amplitude

432 General Theory of Ocean Currents in a Homogeneous Sea

of the stream lines would be reduced, and secondly, the entire wave form would be
displaced so that the bottom waves would coincide more with that of the stream lines.
Both changes depend on the depth of the water, as well as on the current velocity
and wave length of the bottom waves. As long as the expression W/Dt is only a small
fraction the deviations from the previous state remain small, but they become con-
siderable when it approaches or even exceeds 1. Therein r is the time in pendulum
hours (see p. 316) in which the deep current requires to move through the wave length
of a single bottom wave. The values found for this expression from observed data are
relatively large, so that it is probable that bottom waves and stream lines are therefore
closely in phase.

In general, the effects of the three factors are of the same type as before but they are
no longer independent of each other; the topographical and the planetary vortex
effects especially are interrelated in a complex way and disturb each other in extended
oceanic areas during the generation of a uniform deep current. In general, an irregular
bottom topography seems to have a tendency to reduce the velocity of the deep currents.
Deep currents do not then play the dominant role ascribed to them earlier. This is
probably the reason why many results of the earlier theory based on the most simple
assumptions were in good agreement with the observed data, although these assump-
tions were only approximately satisfied in nature. If the topography of the sea bottom
is very irregular the topographical and planetary vortex effects will disturb and some-
times destroy the deep currents, so that essentially there will remain only pure steady
drift currents.

The investigation of the effects of the bottom topography on ocean currents has a
direct connection with the discussion on p. 386, where it was stated that a deflection
of a current cum sole would occur on top of a rising sea bottom and a deflection
contra solem on top of a bottom fall. Without taking friction into account a quantita-
tive estimate of this vortex effect can be made. For an extended bottom wave with a
triangular shape (Fig. 185 ; x-axis at right angles to its crest, >'-axis along its crest),
and assuming a uniform current U in front of the ridge extending throughout the total
water mass (depth of water H) and flowing towards the crest, equation (XIII.29) gives:

?^ dC

-^n =f^ and ^ =0; V^O.

dy 8x

Over this bottom ridge under stationary conditions (duldt = dvjdt = 0) the equations
of motion will be

''fx = -^dy-^''=-^^^-''^-

If the origin of the co-ordinate system is placed at O vertically underneath the highest
point of the ridge, the half-width of which {OA = ^45) is /, and height of which at O
is h, then the depth of water will be

d=cl,^{hll)x,

where the upper sign applies for the forefront side and the lower sign for the rear of
the bottom ridge. The equation of continuity requires the same transport through
every cross-section, that is

UH = u{d^{hll)x].

General Theory of Ocean Currents in a Homogeneous Sea

433

20O"''''^30O

Fig. 185. Topographic influence of a submarine bottom ridge on a current flowing normal

to the longer axis of the obstacle. Lower picture: vertical profile through the bottom ridge

(width, 400 km; height, 200 m; water depth, 4 km; p = 30° N.). Upper picture: stream lines

of the main current (U = 50 cm/sec).

This gives

Over the rise the flow thus is subjected to an acceleration acting along the longer axis
of the ridge with a maximum value of —/A/// above its highest point. This acceleration
gives rise to a curvature of the stream lines cum sole. To the velocity u is added a
transverse velocity v which at a point x = ^ — / (^ is the distance of the point under
consideration from point A) is given by

IHl

whereby /"denotes the cross-section of the bottom surface for the distance from A to ^.
The deflection of the current from the initial x-direction will be vju, and for a small
bottom slope is given with sufficient accuracy by vjU.

The deflection on passing over a bottom ridge is the larger, the smoother the sea,
the higher the ridge and the smaller the velocity U. Since in the ocean U is relatively
small, it can be expected that the bottom topography will have a stronger eff'ect on
the currents. Fig. 185 presents a numerical evaluation of a single case: width of
bottom ridge 400 km, its height 200 m, ocean depth 4000 m and 0 = 30° N. while
U is taken as 50 cm/sec (somewhat high because of the absence of friction in the
current). At the crest of the ridge the deflection will be —37° and in the rear of the
rise at its end —55°. The deflection is of course associated with a corresponding change
in the sea-level; to the normal slope directed along the crest is now added a slope
directed normal to the ridge crest and a corresponding lowering of the sea-level along
the .\--direction. If instead of a single ridge the bottom has a series of ridges and troughs

434 General Theory of Ocean Currents in a Homogeneous Sea

the vortex formation is repeated periodically corresponding to these bottom waves.
Figure 186a shows this case for the Northern Hemisphere ; there is a current curvature
cum sole above the ridges and contra solem above the troughs. If the sea surface has an
overall slope so that already at a larger distance from the ridge a current at right angles
to the ridge is produced then a current field will be formed similar to that shown in

(a)

(b)

Fig. 186. Stream line pattern: (a) for currents crossing a wave-form bottom configuration;
(b) for the crossing of a single bottom ridge (Northern Hemisphere, according to V. Bjerknes

and co-workers).

Fig. 1 86^. The stream lines approach the ridge directly at right angles and pass over
it bending cum sole on the forefront side and contra solem in its rear and then finally
return to their original direction. This latter curvature in the rear can, however, only
occur if there is a convergence on the lee side which is stronger than the divergence on
the forefront side.

Recently, Gortler (1941) has gone into this problem more carefully taking into
consideration the frictional effects also. The mathematical formulation is different as
compared with the previous one and shows an improvement in so far as it leads to
simpler basic equations which are more likely to be solved quantitatively. The results
otherwise agree with those obtained previously. Gortler dealt mainly with a case
similar to that above. The bottom ridge was assumed to have a vertical profile
^ = Po{l + cos (2ttII)x} with|jc| < y and h = 0 outside this region. A horizontal
projection of the stream lines of the main current is shown in Fig. 187 in the same way
as in Fig. 185, but here friction has been considered. For an insight into the frictional
effect the dimensionless quantity hrlH is decisive where hr depends on the frictional
depth and H is the depth of the sea. This quantity usually appears in the expression
G = (Rll)l(hrlH), where R = [///gives the radius of inertia associated with the
current velocity U (equation XIII.26), with which the flow approaches the obstacle.
The different curves in Fig. 1 87 show for a fixed value of Rjl the effect on the course of
the stream lines of the disturbance in the equilibrium between gradient and Coriolis
force above the ridge due to the generation of a "secondary" current. When C is 3

General Theory of Ocean Currents in a Homegeneous Sea

435

or greater there is no essential difference as compared with the frictionless case
(hr = 0, G = oo). For reasonable values of H and / Gortler estimated the magnitude
of G as between 3 and 80, depending on the intensity of U, the latitude and the rough-
ness of the bottom. This shows that for actual conditions in nature everything is the
same as in the case of no frictional influence. This is important for the practical use of
the above results. The effect of the topography of the sea bottom on the course of
the ocean currents has been clearly demonstrated for many oceanic regions. Ekman

-10

Fig. 187. Upper picture: stream line pattern for a crossing of a bottom ridge depending on
friction. Lower picture: vertical cross-section through the bottom ridge.

by using these principles was the first to offer an explanation for the striking
bending of the current trajectories, of the dynamic isobaths south of the Newfound-
land Banks (Helland-Hansen, 1912) which was not understood by simple reasoning.
The course of the stream lines is in good qualitative agreement with that given by
theory for the changes in depth actually present even if a closer qualitative examina-
tion of the phenomenon was not possible.

The dynamic evaluation of the observational data made by the "Meteor" expedition
in the South Atlantic has afforded a good example of these effects of the bottom topo-
graphy (Defant, 1941 b). This example makes it very probable that the large irregulari-
ties in the east-west course of the dynamic isobaths that were found in the western
part of the convergence zone between about 25° and 50° S. have a fixed position and
can be attributed primarily to the morphology of the sea bottom. If the lines of con-
vergence and divergence for this disturbance are traced on transparent paper and laid
over a depth chart the relationship between the two phenomena shows unmistakably.
These conditions are illustrated by a diagram in Fig. 188. The lower part of the figure
shows two depth profiles at 30° and at 35° S. extending from the South American
continent to 0° W. ; they indicate the course of the bottom irregularities running in a
meridional direction as far as the mid-Atlantic Ridge in this part of the South Atlantic.
In the upper part are shown the stream lines plotted according to the dynamic isobaths
over the area from 30° to 45° S. Every "wave trough" in the bottom corresponds to a
bend contra soletn in the stream lines (here the reverse of the conditions as shown in

436

General Theory of Ocean Currents in a Homogeneous Sea

Fig. 186, since this is in the Southern Hemisphere). The extremes do not always coin-
cide in position but particularly in the eastern part are in excellent agreement.

Schumacher (1940, 1943) has indicated further examples. Over the mid- Atlantic
Ridge especially, there is often a corresponding bending of the current to observe.
The large stationary cum sole vortex off the eastern side of the Azores plateau must also

Fig. 188. Upper picture: bottom topography and stream lines for the gradient current in

the disturbance region of the subtropical convergence zone in the South Atlantic Ocean

(30°-45° S., 50-0 W.). Lower picture: vertical bottom profiles at 30° and 35° S. according

to the depth chart of the Atlantic Ocean.

be favoured by the bottom topography. In the Equatorial Counter Current the presence
of the Atlantic Ridge shows this very typical effect. If the water masses are stratified,
there will be corresponding displacements in the isosteres inside the region of influence
of the bottom irregularity (see p. 558). If an isolated submarine ridge lies in the path
of a current a cyclonic vortex will be formed above it. An example of a vortex of this
type is given in the description of oceanic conditions around the "Altair" submarine
volcano in the North Atlantic (Neumann, 1940) (see also, Schott, 1939).

In discussing the effect of the bottom topography on ocean currents it has always
been assumed that the current is more or less uniform from the sea surface down to
the sea bottom. In almost all cases, however, the velocity of the current falls off rapidly
with depth and in addition there are changes in the direction of the current. In these
circumstances it is not so easy to accept a direct effect of the bottom topography on
the current in the upper layers of the sea, since these are often separated from the bottom
currents by very thick motionless water layers or layers with quite a different type of
current. Attention should be drawn to these considerations in any discussion of the
effect of the bottom topography on the currents.

5. Ice Drift

The wind drift of the ice in the polar regions (see pt. I, Chap. VIII, p. 243), like
the ordinary wind-driven ocean currents, is dependent on three forces: wind stress, in-
ternal turbulent friction and Coriolis force; in addition to these it is also affected by

General Theory of Ocean Currents in a Homogeneous Sea 437

a resisting force arising from the random movement of the ice which is proportional to
the drift velocity and acts in the opposite direction. This ice resistance is the reason why
the Ekman theory for the ice drift is inadequate. Nansen had already shown in 1902
from the "Fram" data that the ice resistance cannot be neglected and indicated that one
of its effects must be the small deflection angle observed for the ice drift. Brennecke
(1921) and Sverdrup (1928) have made important contributions to the clarification of
the interrelated forces acting and that of Sverdrup can be regarded as a complete
theory of the ice drift (see also, Rossby and Montgomery, 1935). However, the
observations of the "Fram" are not suitable for testing this theory, since the ice drift
here includes a component due to the permanent surface current (see p. 358), but over
the North Siberian Shelf ("Maud" observations) and in the Weddell Sea ("Deutsch-
land" observations) the ice drift is free from a basic current and is suitable for this pur-
pose. There is, however, one fundamental difference between these two drifts, due to the
very different hydrographic conditions under which these drifts occur, and this has a
considerable effect on the nature of the pure drift current (without ice).

Over the Siberian Continental Shelf the oceanic structure consists of essentially
two layers: a top layer of lighter water and a heavier bottom layer separated by a
sharp density transition layer (thermocline). In the surface layer the vertical equili-
brium state is indifferent (neutral) throughout almost all the year and the turbulence
in it is intense. In the discontinuity layer it falls nearly to zero and this therefore has
the character of a gliding layer. The entire water mass of the top layer is thus drawn
along with the surface current and this, together with the ice masses floating in it,
behaves like an elastic sheet. The resistance against the movement thus arises from
the effect of varying winds driving this sheet together. In the deep Weddell Sea the
oceanographic conditions are different; here there exists no transition layer near to
the sea surface and the density increases continuously with depth. A drift current thus
develops in the normal way, and also the expected decrease in the velocity of the current
and its turn in direction could be observed. In the Weddell Sea it appears necessary to
take into account the effect of turbulent friction besides that of the ice resistance.
These circumstances require to deal with each of the cases separately.

A shallow sea with a density transition layer (thermocline). The wind stress is taken as
proportional to the wind velocity u' and thus as equal to cw (c is termed the wind
effect); the resisting force (ice resistance) as proportional to the velocity of the ice
drift and in opposite direction of it is denoted by —ku (with components — A:m^ and
—kUy along the co-ordinate axes). Then as shown by Sverdrup for the case of the
North Siberian Shelf, for non-accelerated motions (wind along the positive j-axis)

This gives

where

kUx

-/«. = o

and

kUy + /Wj; = CW

n —

cfw

and

ckw

"x -

" k^+p'

Ux f

„— J

u c .

(XIII.60)

tan a = — , „,,v. , — — y.-^
Uy k vv /

438

General Theory of Ocean Currents in a Homogeneous Sea

Here a is the angle of deflection of the ice drift from the wind direction and r is the
wind factor (relative drift velocity, p. 418). Both the angle of deflection and the wind
factor increase with decreasing ice resistance if the wind effect is constant.

It can easily be shown that the end-points of the vectors of the wind factors must lie
on a circle with its centre on a straight line at right angles cum sole to the wind direc-
tion. Its radius is /? = c/2/. In Fig. 189 the vectors shown represent the drifts for
values k ^ Sf, 3/ and/.

Fig.

189. Relation between wind and ice drift for stationary wind conditions and for
diflFerent ice resistance (according to Sverdrup)

A deep sea with a continuous vertical density increase. Here the equations of motion
are the same as for a pure drift current (XIII.23). The boundary conditions are, how-
ever, the following (wind along the positive >'-axis) :

f(u)u^ and ~ -^ = — F(w)w +f{u)Uy

forz = 0:

P S^ P

for z — co: Ux "= Uy = 0.

The functions /(m) and F(h') are for the moment unknown. F(vv)vv is equal to the
wind stress T. With these boundary conditions a solution for the equations is thus

Doj sin </> , u

^r^^ and r = —
w

tan a =

F(vv)sina. (XIII.61)

Doj sin 4> + 71'/(m) vv Doi sin ^

Also in this case the wind factor decreases with increasing ice resistance for otherwise
equal conditions, since the angle of deflection a decreases with increasing resistance.
As in the previous case, the end-points of the relative drift vectors drawn from the
starting point of the wind vector lie on a similar circle as before. The radius is, how-
ever, R = {ttF (w)]l{2Dco sin ^). The functions introduced here are not identical with
the coefficients k and c used in the previous case, but are in a way similar to them.
The function/(M) depends on the state of the ice while F (vv) is related to the turbulence
state of the wind blowing over the ice.

The observations made during the ice drift allow the determination of both a and
r in both cases, and from these the coefficients k and c in the first case and the functions
/(«) and F{w) in the second can be determined. For a test of the relations only those
periods can be used, of course, in which a quasi-stationary state prevails. These
factors are grouped according to increasing wind factor and increasing deflection angle
and presented in Table 1 28 ; Fig, 1 90 shows these mean values in a graphical presentation

General Theory of Ocean Currents in a Homogeneous Sea

439

for a comparison with those required by theory (see Fig. 189). The theoretical relation
is satisfied reasonably well, indeed, but the individual values are strongly scattered —
which in view of the possible sources of error is not surprising. With the wind direction
almost constant the coefficient of the ice resistance k computed from the "Maud"
values decreases from 5-75 to 1-21. In the "Deutschland" values the resistance function

Fig. 190. Observed relation between wind and ice drift for a constant wind influence, but
for an increasing ice resistance.

f{u) increases with increasing wind and drift velocities and in fact so, that a linear
function is obtained for/(w). For the ice resistance this gives

f{u)u = au^.
It is thus approximately proportional to the square of the drift velocity.

For the ice drift over the North Siberian Shelf Sverdrup found that the ice resis-
tance was directly proportional to the drift velocity. This difference can be explained
by the different nature of the ice cover in the two cases. Over the Siberian Shelf the
sea is covered throughout the year by a solid connected ice layer, about 3 m thick
(Pt, I, p. 273). In the Weddell Sea, on the other hand, the ice cover forms only through-
out the winter and also then is not nearly as thick as the Arctic drift ice. Furthermore,
in the Weddell Sea even in the winter there are frequent long open spaces in the ice
cover ("Wacken") so that even at low wind speeds the ice has a much greater freedom
for movement.

Table 130. Relationship between wind and ice drift under quasi-stationary conditions

{mean values)

"Maud"

"Deutschland"

Group 10^ X r

< 1-50

1 •51-200

> 200

102 X r

< 2-8

>2-8

< 30"

3I°-40°

> A0°

< 29°

> 29°

102 X r .

0-77

1-75

2-07

102 X r

2-32

3-39

a ...

13-8°

36-5°

49-3°

21-8°

42-8°

10* X yt

5-75

1-90

1-21

lO^a

150

0-7

10« X c

4-56

4-15

3-86

3-4

440

General Theory of Ocean Currents in a Homegeneous Sea

According to the observational data the wind function F (w) can be approximately
given the form

F{w) = b^/w,

so that the wind stress T = bw^'"^. By this the results of Palmen are brought in mind
because they are in a way similar. The coefficients a and b thus like k and c characterize
the strength of the ice resistance and the effect of the wind.

The seasonal changes in the relationship between wind and ice drift fit in well with
the above considerations. Table 129 shows these changes, together with the calculated
variations in the resistance coefficient and in the wind effect. Over the North Siberian
Shelf both the relative drift velocity and the angle of deflection show a pronounced
minimum in spring and a maximum in summer. This is partly due to the change in
the resistance coefficient k and partly due to the wind-effect c.

The value of k increases gradually from a summer minimum until the first half of
the winter and then rises rapidly to a maximum at the end of the winter in order to
fall off again just as rapidly to the summer minimum. These variations can very well
be explained by the state of the ice cover during the year. In summer the ice resistance
is small due to the numerous open spots ("Wacken") and consequently greater free-
dom of movements for the ice. In autumn and at the beginning of winter these open

Table 129. Seasonal changes in wind factor, angle of deflection, resistance coefficient
and the wind-effect on the ice drift

Jan.-Feb.

Mar.-Apr.

May- June

July-Aug.

Sept.-Oct.

Nov. -Dec.

"Maud"

102 X k

1-67
29-4

1-43*
17-9*

1-67
23 0

2-20
40-8t

2-30t
39-4

1-79
30-8

10* X A:
10«c

2-51
4-82

4-66t
6-97t

3-46
612

1-63*
4-76*

1-72
512

2-37
500

"Deutschland"

102 .^ ^
a°

—

3-21
418

2-23
300

2-90
3-33

2-85
2-48

<3 00)
(2-71)

103 X a

10* X b

—

2-6
3-2

7-8
2-7

6-9

3-2

9-5
3-9

(11.2)
(40)

* Minimum; f Maximum.

stretches are covered with fresh ice, and the ice pressure increases the resistance until
a maximum resistance is reached at the end of the winter when the ice-cover is strongest
and most solid. The annual variation of the wind effect c is more complex. Sverdrup
was, however, able to show that it was in full agreement with the turbulent state of
the air movement over the ice. In the Weddell Sea also the seasonal changes in a and
b are completely analogous. The ice resistance shows, in general, an increase during
winter and spring, but the changes from month to month are more pronounced and
irregular because of the stronger changes in ice conditions of this broken cover. The
coefficient of the wind effect h follows a regular course with the lowest values around
the middle of winter and with an almost steady increase towards the end of winter.

General Theory of Ocean Currents in a Homogeneous Sea 44 1

This also was shown as at least partly dependent on the turbulent state of the air
above the ice.

The ice drift thus to a large extent follows regular laws; it is dependent on three
forces : the effect of the wind on the ice, the frictional resistance between different ice
masses and the dei!ecting force of the Earth rotation. The much greater wind factor
over the Weddell Sea than over the open ocean (see p. 449) is due to the fact that the ex-
posed surface of the ice is more favourable to the action of the wind than that of the
freely moving open sea. Over the Siberian Shelf, on the other hand, the wind factor ob-
served was smaller than over the Weddell Sea; this may be due to the thickness and
compactness of the Arctic ice cover which must offer a much greater resistance to
movement than the ice of the Weddell Sea.

6. Inertia Currents

In the preceding sections ocean currents in a homogeneous sea have everywhere
been considered as stationary phenomena. Observations show that in most cases this
assumption corresponds more or less closely with actual conditions. However, it
can hardly be assumed that the forces involved will always be in equilibrium. Any
disturbance of the equilibrium must, however, alter the state of motion of the water
masses and in this the inertia of the water will play a major role. It is only in more
recent times that one has started to draw attention to such phenomena,

{a) Inertia Currents as Disturbances of a Steady Current

A water mass moving frictionless in a horizontal direction under the action of a
gradient force will, speaking completely in general, be subject to the equations of
motion (X.16). If the .v-axis is taken in the direction of the pressure gradient
(dpjdy = 0), and this pressure gradient corresponds to a steady current (geostrophic
current), then

1 cp
Fo = ^ V- and U^ = 0
fp ex

and one obtains (disregarding frictional forces)

'i/=^^^-^»^ '"^ it = -^"'

A periodic solution for an observer moving with current is

u = t'o sin// -f Uo cos ft,
V = Vq cos ft -f "o sin/r + Fq,
or

u = Co sm {ft -f ip),
V = Co cos (ft + 0) + Vo,
where

^0 = Vi^l + '") ar'<i t^n ^ = —

442 General Theory of Ocean Currents in a Homogeneous Sea

Co is the impulse of disturbance imparted to the steady current Fo at the time r = 0.
If this disturbance is only applied in the direction of the steady current and if at the
time ? = 0 the total velocity is denoted by V, then

M = (J/ - Ko) sin/r and v=Vo-\-{V- V^) cos//. (XIII.62)

If the permanent equilibrium of a steady current is disturbed, the difference between
the disturbance vector and the steady gradient current is transformed into an inertia
movement with a corresponding circle of inertia. The period of the circular movement
is

T = -77 = — -. — ; = I pendulum day.
/ oj sm <^ ^ *^

The amplitude of the two velocity components is the same, and the phase of the
>'-component precedes that in the .v-component by one-quarter of a period. These are
the characteristic features of a pure inertia movement. It is superimposed on the uni-
form gradient current and thus gives an oscillating flow, the period of which depends on
the Coriolis force. This period is identical with the period of one revolution around the
circle of inertia; numerical values for it are given in Table 1 12a (see p. 316) for differ-
ent latitudes. Inertia oscillations are not associated with any large transverse displace-
ments of the water masses, since the disturbance velocity c = V — Vq usually remains
small. The magnitude of these can be taken from Table 2 for different latitudes and
velocities. In the open ocean these transverse displacements are usually of little
importance but they are still characteristic phenomena which are quite noticeable in
current measurements.

If pressure forces are present in a homogeneous sea due to a slope in the sea surface
{dijdx = 4; dijdy = iy) the equation of motion (XIII.3) will apply. A steady motion
(geostrophic current) is associated with a corresponding slope of the sea surface given by
/j. and iy so that

- / - /

^x=- V and 'V = - ^ ^•

If, further at the time / = 0, there are current components Mq and i\ and slopes
ij, 0 and iy o which do not correspond to the condition for a steady state, then the above
equations have the following general solution (Fr. Defant, 1940):

w = t/ + ("o - U) cos ft + [vo - (glf)ix,o] sin//,
V = V-\-{vo- V) cos ft - [uq - (g//)/x_o] sin//,
ix = ix + ['x,o — 'x] cos ft — [iyo — iy] sin ft,

iy = h + [iy 0 — Iy] COS ft + [/^ „ " I x] siu//.

(XIII.63)

This set of equations shows that for a completely free initial state, both the current
field and the sea surface will perform inertia oscillations around their equilibrium
position which, however, will not correspond in all points to the conditions for pure
inertia waves. In the current field the amplitudes of the corresponding velocity
components will be equal only when the sea surface slope corresponds initially to the
steady state. But according to the second pair of equations the sea surface does not

General Theory of Ocean Currents in a Homogeneous Sea 443

perform any inertia oscillations at all but will rather remain from the beginning in
the stationary position. Thus in the general case the amplitudes of corresponding
terms are no longer equal and the motion is then elliptical instead of circular. However,
the amplitudes of corresponding terms in the sea surface oscillations are always equal
and these are therefore always pure inertia movements. It has been found that currents
flowing into a wide area uninfluenced by coastal eff"ects usually follow a wave-form
course rather than a straight course. A current with an oscillatory streamline seems to
be a more stable type of motion than one with a linear course. Once a bulge is formed
in any direction, the centrifugal force draws the water further and further out and the
bulge produced by such disturbances will grow steadily; consequently, progressive
waves and vortices will be formed in which the current will oscillate about a mean
direction. In dealing with problems concerning these oscillating currents it is of course
necessary to take the Coriolis force into account (Exner, 1919). It surpasses the scope
of this section to penetrate more deeply into the dynamics of progressive waves of this
type in an infinitely extended medium ; it rather belongs to and will also be discussed
when dealing with the theory of progressive tidal waves (Vol. II) ; for an account of
progressive waves with inertia period see Fr. Defant (1940) and Ekman (1941).

{b) Inertia Movements Associated with Drift and Gradient Currents

In the formation of steady drift and gradient currents the state of motion changes
from the first motionless initial equilibrium state into a second state in which there is
an equilibrium between all the forces acting. It can be expected that this transfer will
give rise to inertia oscillations which will gradually be damped by friction until the
new stationary equilibrium state is reached. Ekman (1905) examined in some detail
the case of a suddenly starting wind over a deep, extended ocean. A comprehensive
treatment of all questions arising has been given by Fjeldstad (1930). The equations
of motion (X.16) which stand in question can be combined introducing u -f- iv = w
(/ = \/—\)'m order to obtain a single equation

dw T) 8^w

J, + '>• = I J?- (^"'-^^^

The boundary conditions to be satisfied are

for t =Q: vv = 0

and / , dw ^ ^

for all t:

If the wind arises suddenly at a time / = 0 with a tangential pressure Tin the direction
of the positive j-axis, then the velocity components u and v are given by

IttT f^ sin Itt^

and

V =

pDf Jo V^ "^ \ ^D^

IttT f"^ cos 2tt^

pDf .

exp -T7^,U^ (XIII.65)

Vi ^^Pi-4^^i^^

444

General Theory of Ocean Currents in a Homogeneous Sea

D is the frictional depth and r = {ftjl-n) is the time expressed in units of 12 pendulum
hours. The gradual formation of the drift current can be illustrated by plotting the
time-variable velocity vector at different depths in the form of the hodograph curves
given by Ekman (Fig. 191). For a suddenly starting wind this curve at the surface will
have the form of a damped circular oscillation with a period of 12 pendulum hours
which is superimposed on the final stationary state of motion. In the deeper layers the
oscillation will at first grow somewhat and then regularly decrease again. The velocity

^=0

Z = 05D

z^D

Z-- 2D
1^ >

Fig. 191. Hodograph curves for different depths of the pure drift current which develops
due to a wind beginning suddenly (ocean depth unlimited).

components can also be plotted separately along the time axis thereby obtaining the
curves shown in Fig. 192. Each component shows a damped oscillation around a
stationary final state and both together show very clearly the characteristics of inertia
oscillations. At different depths the oscillations have exactly the same phase but with
a decreasing amplitude. If the water depth is less than the frictional depth the close
distance from the bottom becomes apparent in the curves, but the oscillation is strongly
damped only in the immediate vicinity of the bottom; for hjD = 0-25 the current
approaches almost aperiodic the steady state. Solutions can also be found for the case
in which the effect of the wind is not applied suddenly but only gradually, and also
for the case where the wind maintaining a wind drift current either suddenly or
gradually ceases. For more detail see Hidaka (1933), Nomitsu (1933), Fr. Defant
(1940).

The sudden formation of a sea surface slope in a similar way as in the case of a
drift current must also give rise for a gradient current to inertia oscillations. Ekman
has given the theoretical basis also for this case and has pointed out that for an ocean
of greater depth, due to the small frictional effect in the geostrophic current, these
inertia oscillations will die away very slowly so that a longer duration of these must

General Theory of Ocean Currents in a Homegeneous Sea

445

be assumed. If the pressure gradient, due to a suddenly imposed sea surface slope, acts
along the positive j-axis there will be an extra term +/(/ (to be added on the right-
hand side) in the equation of motion (XIII. 64), where U is the velocity of the steady
gradient current (geostrophic current) corresponding to the sea surface slope. This
equation must be solved assuming the boundary conditions that for z = 0 : dwjdz = 0
and for z = h\w = 0 and for r = 0 : u- = 0 and for r = oo : iv = U (stationary state) ;

0-75

0-50

0-25

0 00
0-75

050

0-25

000
025

ooo

-0-25
0-25

000
-025

Surface

N and E components inunits TDlWu

1 1 1 • i \ ^

' / *■

/'"/

/' }

i 1

n'\

.>H

^'^ ^<-^/

-■^

-X

-<:x

/' '^r\ ' '

/ l\ \ ''V^\

1 ^~~-

' / \ \ / / "^ V !'- y

f*^ ^ ~^\.

h/n =li

1 /E n\i Vi /

" ->C

Z--0

; /

-' T

f ^

^ — ^ I 1

N^\

\^__

-'-'--

k.i--l-— ^--

^ 1 : '

— 1 — 1

..1

1 ; ' \ ■ '

0 4 8 0 4

Pendulum hours

Fig. 192. Upper pair of curves: drift current of an ocean of infinite depth for r = 0 (surface).
Lower pair of curves: drift current for hID = 1| and in fact for z = 0 (surface), z//z = 0-3
and zjh = 0-6 (north and east components always in units TD/nfj. according to Fr. Defant).

the velocity components of the steady gradient current are denoted by Ust and Vst-
Introducing again u + iv = w, then the equations of motion reduce to the single
relation

For stationary conditions (Bwldt = 0, equation XIII. 30) the solution is given by
equation (XIII. 31). Under non-stationary conditions a solution is obtained most easily
by assuming

»*• = n',< - H's<F(0

with the condition

/■(od) = 0 for ?= 00.

This can only be given as a series which, however, converges rapidly. As in the case
of developing gradient current, oscillations about the final stationary state with inertia
periods and decreasing amplitude are produced in both components.

446

General Theory of Ocean Currents in a Homogeneous Sea

For the upper layers when the depth of the sea is great, one obtains in close approxi-
mation

u=U

U cos ft and v=Usmft, (XIII.66)

which indicates a simple harmonic inertia motion with an amplitude U. In reality
of course a decrease always occurs in a spiral form but when the depth is large this
decrease is extremely small. These oscillations can be found by observation as far
down as near to the sea bottom. Figure 1 93 shows the course in the velocity components
at the surface, for a middle depth and for a layer near to the bottom in the case where

-10
20

10
0

k- r^

„^' "

.-V-J
\

^^\

,_/

/7n

L Z/h--Q5
\ /

v^/V

:^'^

^>^_.

-.^

•

-'-''

^ ,

Z//>=09

;>^-

"^^"^•9^

h-f-

i~~

T'"

'■

1 ~-

--K"

.__,- T

— '

8 0 4?

Pendulum, hr

8 0

Fig. 193. Gradient current in the ocean for an ocean depth hlD = -j, and in fact for z = 0
(surface), z/A = 0-5 and 0-9. (Values for the north and east components for «/C/ and vllJ,

according to Fr. Defant).

h\D = f . It can be seen that due to small damping the amplitude of the inertia
oscillation is still quite large in the mid-depth. Calculations can also be made for the
case of a gradually developing sea surface slope; the amplitude of the inertia oscilla-
tion produced depends on the rate at which this slope develops but the character of the
oscillation is still kept.

The preceding discussion leads to the expectation that inertia currents will be of
frequent occurrence in ocean currents, but a considerable time passed before their
existence was actually proved. This was due to the circumstance that in order to prove
the presence of cum sole, turning current variations of this type corresponding to their
period, current measurements from an anchored ship over an interval of several days
were needed. Measurements of this type are only seldom made and are associated
with considerable difficulties which have been overcome only in recent times. The
first current measurements in which the presence of inertia currents was suspected,
was the long series of measurements made by Helland-Hansen and Ekman (1931)
in the trade wind region of the Eastern North Atlantic. At the anchor station with the
longest observational period (141 h, 30-2° N., 14-0° W.) there were, besides oscillations
with tidal periods and also others with inertial oscillation, periods with 23-844 mean

General Theory of Ocean Currents in a Homogeneous Sea

447

hours. This period is only 13 min shorter than the diurnal moon period K^ of 23-94 h
which presumably was also present. This difference is, however, sufficiently large to
decide out of six wave trains which of the waves is present. Figure 194 shows the diurnal
regular oscillation after elimination of the semi-diurnal tide. While during the first
three days there was a regular damping of the waves, at the end of the series there

0 12

9-2tt 1930

torn

II-2IL

l2-5nL

135m:

Fig. 194. North Atlantic Ocean: anchor station of the "Armauer Hansen" 30° 13' N.,
13° 57' W. Current measurements in 5 m depth after elimination of the semi-diurnal tide.
Full line, north component; dashed line, east component; velocity scale in mm/sec. At the
upper rim moon hours. The distance between two vertical lines is very nearly 6 pendulum
hours (according to Helland-Hansen and Ekman).

appeared to be a phase shift in the meridional component due to a new disturbance ;
the oscillations then lose rapidly in regularity. Harmonic analysis for the first three
and then for the following three days gave (cm/sec, / in pendulum hours) :

cos (27r/12. t - 112^) , N = 1-58 cos (27r/12. t - 102")
115^) ^"^E

N = l

E = 1-51 sin (27r/12. t

1-29 sin (277/12. / - 97°)

These oscillations are pictured by the full and dotted sine curves in Fig. 194. The
good agreement led Helland-Hansen and Ekman to interpret these waves as inertia
movements. The phase difference between the two components was 12 min more for
the first days and for the second three days 20 min less than the theoretical required
value of 6 h. The average ratio of the amplitudes was 1-23 as compared with a
theoretical value of 1 . The oscillations were thus of the elliptic type with a ratio of
5 : 4. Considering that besides the inertia oscillations presumably the diurnal tide was
also present, the results obtained are very satisfactory.

An unambiguous proof of the occurrence of inertia oscillations was provided by the
current measurements organized by H. Pettersson in the Baltic. As an adjacent sea with-
out any significant tides this is particularly suitable for such an investigation. Gustaf-
SON and Otterstedt (1932) and Gustafson and Kullenberg (1933, 1936) have
made a detailed analysis of the suitable current measurements in the Baltic; in many

448

General Theory of Ocean Currents in a Homogeneous Sea

cases there was no doubt of the presence of pure inertia movements. The best example
is that contained in the measurements of 17-24 August. The recordings were made
between Gotland and the mainland (57-8° N. 17-8° E., depth 100 m) at a depth of
14 m over a period of 162 h. The structure of the sea showed a well-developed density
transition layer (thermocline) at 25 m and an almost homogeneous top layer. The
variations in direction and strength of the current can be given in the form of a pro-
gressive vector diagram which shows the track of a single small water element.
On the current directed towards the NNW there is superimposed an oscillation
rotating to the right, at first increasing and then decreasing (Fig. 195). The changes

Fig. 195. Inertia oscillations in the Baltic in hodograph representation (according to

Gustafson and Kullenberg).

with time in the two velocity components are shown in Fig. 196. This diagram is
particularly reminiscent of the theoretically derived oscillation due to a suddenly
starting wind or to a suddenly imposed pressure gradient (Figs. 192, 193). If the first
waves of the excitation period are omitted the period of the oscillations is 14-0 h as
compared with 14-14 h for the inertia oscillation. The phase difference is almost
exactly a quarter of a period and the amplitudes are very nearly equal.

The meteorological observations taken at the same time do not permit any deduc-
tions about the origin of this inertia wave. The question concerning the horizontal

General Theory of Ocean Currents in a Homogeneous Sea

449

extent of this type of inertia oscillation was examined in later current measurements in
the Baltic. Recordings from four anchored oceanographic research vessels between
the Latvian coast and Oland (along the 56° 20' parallel) in the Baltic all, except for
the vessel next to the Latvian coast, showed regular inertia oscillations at 15 m depth
(density transition layer) with amplitudes of up to 20 cm/sec. The inertia oscillations
(period 14-42 h = i pendulum day) had almost the same phase value but decreased
very rapidly towards the coast. The water masses along the parallel investigated thus
took part as a whole in the inertia oscillations (Kullenberg and Hela, 1942).

Fig. 196. Velocity components of the currents pictured in Fig. 195 (according to Ekman).

Specific inertia oscillations were found at the "Altair" anchor station in the area of
the Gulf Stream north of the Azores (44-6° N. 34-0° W., 16-20 June 1938). Analysis
of current measurements made at this station down to great depths (Defant, 1940 b)
showed that besides the semi-diurnal tide there was also a 17 h oscillation; this had a
large amplitude at all depths but the phases changed with depth. These phase changes
which are related to the oceanographic structure at this station indicate that these
inertia oscillations were coupled with internal waves which are the expression of a whole
system of inertia oscillations of the surrounding water masses (see Vol. II). The com-
bination of the 1 7 h wave with the semi-diurnal tide gives rise to beat phenomena with
a period of 14-3 h and a beat interval of 1-86 days. This shows in a typical way the
current values at all depths so that there can be no doubt that besides the tides,
inertia oscillations were present here. Harmonic analysis gave besides the tidal wave
also the value for the 17 h wave presented in Table 130. Division into different layers
follows from the similarity in the phase which between 15 and 30 m and between
500 and 800 m shows an abrupt change of about half a period. The 1 7 h wave shows

Table 130. Inertia oscillations at the ''Altair' station
(16-20 Jime 1938; 44-6= N., 34-0° W.). Period 17 h

Depth of
layer
(m)

Current

Ratio of
ampl.

N.:E.

Phase

N. + 4-25 /;

A^-component

E-com

ponent

Difference

A-mpl. Phase 1 Ampl.
(cm, sec) (h) (cm sec)

Phase
(h)

5-10

30-100

300-500

800

8-75 14-45

100 7-33

50 9-0

80 20

8-25
1 0-0
50
80

1-85
12-27
14-85

106
1-00
1-00
100

1-70
11-58
13-25

6-25

+015
+0-69
+ 1-60

+ 1-75

450

General Theory of Ocean Currents in a Homegeneous Sea

all the characteristics of inertia oscillations. At </> = 44° 33' the theoretical period is
17-1 h. The amplitudes of the components are almost identical and also the require-
ment that the ^-component should follow the TV-component by a quarter of a period
(4-24 h) is fully justified. The deviations in the deeper layers must be a consequence of
internal waves. Figure 197 shows the course of the N- and ^-components for the layer

,N E
*zo -no

*io 0

0-10

-10 -20

-20-30

17.2. M.G. Z.
6 12

laa.

io.

0 6

2asi

18 0 6 12

Fig. 197. North and east component of the current at the "Altair" anchor station according
to the values of the harmonic analysis for the depth interval 5-15 m (basic current + 17 h

period + 12-3 h period).

1 1

■1 1

1 1

I 1

I 1 ■

-T 1

I i

1 1

I 1

II 1 ;

'"\

/"■

E ,

/
1
1 /

--^

.-vC

^ /
\ /

\ \

/
/ /

— ^.,^

rt-~^

'\ \

1 1

^x/^^-

sV/

■^

1 1

1 I

1 1

1 ,1

1 I

1 1

II 1 i

1 1

between 5 and 15 m as given by the values obtained by harmonic analysis. The beats
stand out clearly, as does the retardation of the £"-component behind the TV-component
characteristic of inertia oscillations.

The inertia oscillations at the "Altair" anchor station are of considerable interest
in so far as they show that the entire current system, together with the oceanic structure
of the surrounding waters, seems to take part in these oscillations following the
rhythm of the inertia period. These oscillations which may be initiated by any external
disturbance impulses stand out particularly well in stratified waters, since coupled
with these oscillations of the flow are corresponding oscillations of the density transi-
tion layer and of the system of isosteres which are thus reflected in all layers (see
Vol II).

Chapter XIV

Water Bodies and Stationary Current
Conditions at Boundary Surfaces

1. Water Bodies and the Boundary Surface Between Them

The theory of ocean currents in a homogeneous sea gives results which allow in many
cases its application to actual conditions, although the sea itself is far from being homo-
geneous. In changing from a homogeneous to a stratified ocean it is necessary to
consider two homogeneous water masses (water bodies) situated side by side and
separated by a discontinuity surface (boundary surface). On passing through this,
changes in physical and chemical properties occur and also in the state of motion of
water masses. This is, of course, also only a schematic model, since in reality the indi-
vidual water bodies are not quite homogeneous and the transition from one to the
other is seldom abrupt. Usually in Nature there is a rapid "transition layer" between
the more or less homogeneous water bodies inside which a steady, rapid change of the
properties occurs, while passing through it.

The genesis of boundary surfaces of this type is due to the circumstance that in
certain oceanic regions specific water types are continuously formed and carried away
by the ocean currents together with their characteristic properties. In this way two
different water bodies are brought into close contact at singularities in the current
field and a boundary surface between them is formed at convergence lines. The prin-
cipal changes in the horizontal distribution of a property (such as temperature or
salinity and others) occur always in connection with so-called deformation fields of
the motion (Bjerknes and co-workers, 1933). The most simple case of a horizontal
deformation field is the current field at a neutral point (see p. 365, Fig. 155 a) with
hyperbolic stream lines in each of the sectors formed by intersection of the two stream
lines in the neutral point (Fig. 198). These straight lines are the principal axes of defor-
mation of the field; one of them is an axis of dilatation and the other at right angles to
it is an axis of contraction. This deformation field when superimposed on the field of
one of the water properties will have a marked effect on the latter. The two full lines
in Fig. 198 represent two isolines of a property, such as for example, the temperature.
The current field will produce displacements in the position of these lines: all isolines
which initially are parallel to the axis of contraction will move away from it and
isolines parallel to the dilatation axis will move towards it. It can also be shown that
two isolines through the current will always tend towards a direction parallel to the
dilatation axis, so that they will first move away from each other to a maximum
distance, and then after reaching a certain angle to the dilatation axis will move to-
wards it again. In the case of a temperature distribution the effect of the deformation

451

452 Water Bodies and Stationary Current Conditions at Boundary Surfaces

field is thus a concentration of the isotherms parallel to the dilatation axis and the
horizontal temperature gradient will increase very strongly (according to theory
indefinitely). This, therefore, leads to the formation of discontinuity surfaces the inter-
sections of which with the sea surface show as discontinuity lines or fronts.

Fig. 198. Deformation field and the change of a field of a characteristic water property.
a — a, shrinking axis; b — b, axis of dilatation; 1 — 2, isolines of the property in the begin-
ning; r — 2', isolines of the property at the end of deformation.

The formation of strong horizontal gradients in the boundary regions of water
bodies actually occurs most often in association with stationary oceanic deformation
fields. However, other circumstances are involved in their maintenance. These are
coupled with the effect of the deformation field and may lead to stationary fronts which
are particularly characteristic for the horizontal distribution of the oceanographic
factors. For an initially meridional temperature gradient and a steady meridional
ocean current v the conditions will develop along the following lines (see Pt. I, p. Ill):
the temperature &■ at a fixed point will change according to the relation (positive
>-axis directed polewards)

d^ 1 d^

If V is directed towards the pole, the temperature at a fixed point will increase since
dd'jdy is negative (temperature increase by advection), that is, the isotherms will
be displaced towards the pole provided that Q is small. However, due finally to
the increase of temperature the first term on the right-hand side will also be increased
and as a result all the factors aff'ecting the temperature will maintain an equilibrium
state. Although the ocean current is directed towards the pole the temperature
distribution will remain stationary. Similar reasoning will also apply to the hori-
zontal distribution of other oceanographic factors. Besides stationary fronts of
this type there are also frontal formations due to aperiodic occurring processes.
However, due to the lack of synoptic observations the course of these usually can-
not be traced. An interesting case has been given in Pt. I, p. 182 in the discussion

Water Bodies and Stationary Current Conditions at Boundary Surfaces 453

on mixing processes in the transitional area between the North Sea and the BaUic.
These are real, progressing hydrographic fronts that are associated with the inter-
change of water between the two seas.

The parts of the ocean where more or less stationary fronts are found are actually
closely connected with the occurrence of quasi-stationary deformation fields with an
axis of dilatation in the current system of the oceans deviating as little as possible
from the east-west direction. The position of these can be found directly from a chart
of ocean currents. In the Northern Hemisphere the most important are :

(1) The North At/antic Polar Front which is present with its main part to the south
of Newfoundland and there forms the boundary between the Gulf Stream water and
the Arctic water of the Labrador Current; its continuation separates the cold low-
saline water of the East — and in part also of the West — Greenland Current from the
Atlantic water masses. Other parts lie south of Spitzbergen and in the Barents Sea.

(2) The North Pacific Polar Front with its main part between the Kuroshio and the
Oyashio which can be traced to about the middle of the ocean. These fronts are a
consequence of quasi-stationary deformation fields in the current system in this part
of the ocean.

(3) This is also the case, though less clearly, in the Southern Hemisphere Polar
Front which runs right around the Earth. It lies between the West Wind Drift and the
Antarctic Current. In the parts where it is particularly well developed (for instance,
south of South America and between the Falkland Islands and South Georgia) the
connection with the local deformation field is clearly shown.

2. Stable Discontinuity Surfaces

If two motionless water bodies are present together in the ocean for a stable equili-
brium, the heavier water type must lie underneath of the lighter and the discontinuity
surface between them must coincide with a level surface. Two water bodies at rest,
situated side by side, will never be in equilibrium, even if each water body by itself
has a stable vertical stratification. Since, due to their different densities, the pressure
in each water mass will increase with depth at diff"erent rates, pressure differences are
created; the resultant water movements will overturn the water bodies and they will
only cease when the water bodies are again situated one above the other, separated
by a horizontal boundary surface. However, two water bodies side by side can be in
stable equilibrium // they are in motion. The form and position of the resulting dis-
continuity surface was first given by Margules (1906) following up an investigation
by Helmholtz (1888); a more general representation was given later by J. Bjerknes
(1921) and later an application to the analogous conditions ia oceanic water bodies
has been given by Defant (1929 b).

A stationary state of the boundary surface is possible only for a certain definite
state of motion in the two water bodies; thereby the boundary surfaces will lie at an
angle to the level surfaces, so that the denser water always spreads out in a wedge-
form underneath the lighter water. It will be a discontinuity surface for density
(temperature, salinity or both) but not for pressure, since otherwise movements
would immediately start directed towards the boundary surface. This, however,
would interfere with the condition of stationary state. On the contrary, the boundary
surface will be a discontinuity surface for the pressure gradient. According to the

454 Water Bodies and Stationary Current Conditions at Boundary Surfaces

Hadamard classification (1903)t it is thus a discontinuity surface of zero order for the
density and of the first order for the pressure. The horizontal movements in each
water body must thus be parallel to the boundary surface since otherwise the surface
could not remain at rest.

There are kinematic and dynamic boundary conditions that must be satisfied at
the discontinuity surface (see p. 324). The kinematic condition (equation X.29) re-
quires that

(ill — iio) cos (nx) + (vi — V2) cos (ny) + (m'i — u'a) cos (nz) = 0, (XIV. 1)

where /; is the direction of the normal to the boundary surface ; u^, v^, w^ are the velocity
components of the lighter and U2, v^, Wo are those for the heavier water type. The
dynamic condition (equation X.29) requires that the pressure should be the same on
both sides of the boundary surface (pressure equal counter pressure)

P^-P2== 0. (XIV.2)

If w, V, vv are the total acceleration components and X, Y, Z are the components of the
forces, the equation of motion for the lighter water body 1 can then be written in the
form:

dPi = Pi [(A^i - it,) ^x + ( n - i\) dy + (Zi - vi-i)] dz. (XIV.3)

An analogous equation will apply for the heavier water body 2. The equations

dpi = 0 and dp2 — 0

will then give the equations for the isobaric surfaces according to the motion in each
water body while the dynamic condition (XIV.2) will give the equation of the boundary
surface

[(Pi ^1 - P2 ^2) — (Pi wi — p2 W2)] dx + [(pi Ti — P2 Y^ — (pi Vi - P2V2)] dy +

[(Pi Zi - P2 Z2) - (pi vvi - P2 vva)] dz = 0. (XIV.5)

In the most general form these are the equations for the slope of the isobaric surfaces
in each of the water bodies and for the inclination of the boundary surface.

If the water bodies, each in itself, are both homogeneous (pi and pn = const.), the
motion is non-accelerated {ii — v — w = Q) and is directed straight along the y-axis
(ui = 112 — 0 and \\\ = H'2 = 0), then there will be a static equilibrium in each water
body and

^i=/''i, -^1 = ^ and X2=fv2, Zg = g.

Further, if the slope of the isobaric surfaces in the (.vz)-plane is denoted by
dz\dx == tan ^ and that of the boundary surface by dz\dy = tan y, then the above
equations will give

f f

tan ^^= --^vy; tan /Sg = - - V2, (XIV. 6)

t According to the classification of such surfaces introduced by Hadamard (1903), a discontinuity
surface at which the velocity and the density (temperature and salinity) change abruptly by a finite
amount from one to the other side, is termed a discontinuity surface of zero order. It is defined to be of
Ihe first order when the characteristic properties of the water bodies at the surface change continuously
but their derivatives normal to the surface are subject to abrupt changes.

Water Bodies and Stationary Current Conditions at Boundary Surfaces 455

and

tan y

f P2V2 — PiVi

(X1V.7)

g P2— Pi

In each water body there will be a gradient current (geostrophic current). The angles
/Si and i3o will determine the slope of the planar isobaric surfaces and that of the
physical sea level.

The slope of the boundary surface is of quite a different order of magnitude.
Taking </. = 45° N.; a^ = 28-13; ag = 27-33 (density at 0°C and 35 %o, as well as
at 0°C and 34%o) and in addition if the water body 2 is at rest (t'a = 0), while for water
body I Vi= 100 cm/sec, then one obtains y = 0°46' 13". The boundary surface is
only little inclined to the level surfaces and rises only 13-5 m/km. In the water body at
rest the isobaric surfaces are horizontal; in the upper, moving water body they rise
very slightly to the right of the current direction because ^i = 0° 0' 2-2", which means
a rise of 1 cm in 1 km (Fig. 199; the slopes, in order to make them visible at all, are
shown with a considerable vertical exaggeration). The slope of the boundary surface

i^X

-y

Fig. 199. Stationary current system of two water masses situated side by side (position of
the boundary surface, isobaric surfaces and the physical sea surface); GF gradient force;

CF Coriolis force.

is about 1000 times greater by magnitude than that of the isobaric surfaces and that
of the physical sea level, in the moving water body. Table 131 gives the slopes when
Pi = 1-027; P2 = 1-028; v.^ = 0, for different values of ^i at 45 °N. The lighter water
mass always glides as a pointed wedge on top of the heavier and superimposes the
heavier near the boundary surface as a quite shallow layer.

Equation (XIV.7) can be simplified if the slope of the isobaric surfaces is neglected
by comparison with the much greater slope of the boundary surface. This gives

tan y = —

S P2

(V2 - Vi).

(XIV. 8)

456 Water Bodies and Stationary Current Conditions at Boundary Surfaces
When To = 0 it follows

tany

tan/3i

P2 — Pi

and since pa ~ Pi is of the order of 10~^, the slope of the boundary surface will be
about 1000 times greater than that of the isobaric surfaces; it has, moreover, the
reverse inclination as compared with that of the isobaric surfaces in the upper water
body; the physical sea level has thus the opposite inclination in comparison to that of
the boundary surface underneath.

Table 131. Slope of the boundary surface and the isobaric surfaces for moving water
masses. <^ = 45° N. pi = 1-027, p^ = 1-028; ^3 = 0, ^Sg = 0

V (cm/sec)

0°3'42"

7' 26"

irs"

14'51"

18' 33'

tan 7 ..... .

1:926

1:463

1:309

1:232

1:185

p- ( \ /over 10 km .

Kise(m) "j^ over 50 nautical miles .

10-8

21-6

32-4

43-2

540

100

200

300

400

500

-0°0'0-3"

-0'0-4"

-O'O-?'

-0'0-9"

-0' ir

tan /3i 10-« X ....

105

210

3-15

4-20

5-25

/over 10 km .
^'^"^<*^'"^\over 50 nautical miles .

105

210

315

4-20

5-25

The slope of the Margules boundary surface can also be derived quite readily from the equations
of motion. This will be given here since it will be required later. We consider two water bodies
1 and 2, one above the other, the upper limit of the lower being the boundary surface and the upper
limit of the lighter above it being the physical sea level ; furthermore, we allow only slopes along the
X-axis. The position of the two boundary surfaces can be defined by the deviations ii and i^ from
their equilibrium position at rest (level surfaces). The pressures at an arbitrary point A in the water
body 1 and at a similar point B in the water body 2 will then be:

P\ = (fh + h - =)Pig - Pig^i
and

P2= - Plg^l + (f'l + QPlg + Vh - ^2 - 2)p2g-

Then for stationary state the equations of motion will take the form

fvr+g'^

= 0; 2 fv^ +

■?■ +

P2 ^X

The first equation gives immediately the slope of the physical sea level

tan i3i =

^^1

Elimination of c^j/?x from the second gives the slope of the boundary surface

tan y = £^2 = - / ^^^'^ ~ Pi^i

^X g P2- Pi

which are the same equations as before.

It might be mentioned here that equation (Xrv.7) can also be written

tan y = Pztan^o - Pitan^i

P2 - Pi

which gives the slope of the boundary surface directly from the slopes of the isobaric surfaces. Further
the equations of motion give a relationship between the horizontal pressure gradients on either side
of the surface

8p2 _ 8pi

dx dx

g(p2 - Pi) tan y.

Water Bodies and Stationary Current Conditions at Boundary Surfaces 457

According to equation (XIV. 7), stable boundary surfaces on the rotating earth are
usually inclined and are horizontal only when the specific momentum pv (velocity
impulse) is the same in both water bodies. When/= 0, that is at the equator, discon-
tinuity surfaces are of course always horizontal. The following rule can be deduced
governing the inclination of the boundary surface and of the physical sea level for
steady frictionless currents in the Northern Hemisphere : In every water body there will
be a geostrophic current ; looking in the direction of the current (downstream) the
isobaric surfaces and the physical sea level will rise from left to right. The lighter water
body will be situated on top of the heavier as a very sharp wedge and will move to the
right relative to the heavier when looking from the heavier towards the lighter. In
the Southern Hemisphere this will, of course, be reversed (it is simply necessary to
replace "right" by "left").

A good example of steady current conditions in the simplest form is found inside the
current system of the East Greenland Current. Here a cold low-saline water mass
flows along the coast towards the south; on its left-hand side it borders against the
almost stationary Atlantic Water in the middle part of the European North Sea.
The main core of the East Greenland Current keeps to the west along the shelf of the
east coast of Greenland. Figure 200 shows a density section across the current according
to the observations of the "Belgica" expedition (Amundsen). In the vicinity of the
current the isopycnals rise with a mean gradient of 1 :300 towards ESE. The water
of this cold low-saline current is strongly stratified and especially at the surface there
is a strongly heated, very light top layer. The isopycnal, o- = 28-0, indicates the boundary

Fig. 200. Density cross-section normal to the East Greenland current according to the

observations of the "Belgica" expedition and the observations of Amundsen. (The small

map contains the position of the cross-section and the stations used.)

458 Water Bodies and Stationary Current Conditions at Boundary Surfaces

between the Greenland Current water and the almost homogeneous Atlantic Water
(a = 28-1). The wedge-shaped spreading of lighter polar water over the heavier
Atlantic Water to the east stands clearly out. Taking a^ — 21 -X for the polar water
and a velocity i\ of about —25 cm/sec (towards the south) and a^ = 28-1,
Ta = —5 cm/sec (towards the south) for the Atlantic Water, then equations (XIV.6
and 7) give the boundary surface slope as y = 0° 10' 2" which corresponds to 1 :343
rising towards the east and for the slope of the physical sea level and that of the
isobaric surfaces in the Greenland Current one obtains ^S^ = 0° 0' 0-7" which is about
35 cm in 100 km towards the west. The slope calculated for the boundary surface is in
good agreement with that actually found. The rise of the physical sea level towards
the coast is rather remarkable and even these simplified assumptions lead to the con-
clusion that the sea level along the east coast of Greenland will be on the average about
20-30 cm higher than in the central parts of the Norwegian Sea.

3. Stable Stratification of Water Masses

Water bodies are frequently found in the ocean, situated in a remarkable way side
by side, which are apparently in stable equilibrium. This can only occur if certain
definite current conditions are present in each water mass. The resulting upwelling
and sinking water movements in these water masses must be counter balanced by the
current system present. These conditions take a simple form, if one considers at first
water bodies arranged in strips which are motionless and are embedded in moving
adjacent water masses of a different type (Defant, 1929 b).

(a) A Motionless Heavy Water Body Embedded into Moving Light Water Masses

The conditions required for stationary equilibrium are shown schematically in
Fig. 201 (Northern Hemisphere; reversed current directions in the Southern Hemi-
sphere). This is readily understood on the basis of the rule given above. In the heavier

Heavier water

Fig. 201. Motionless heavy water mass embedded in moving lighter water (Northern

Hemisphere).

water body the pressure at the same level must be lower than in the surrounding water
and correspondingly the physical sea level will be lower than on either side. An
elongated depression of it will thus indicate on the sea surface the position of the
heavier water body which extends in wedge-form in the deeper layers underneath the
moving water masses to either side. If the water body in the middle between the moving
water masses is not motionless then this movement must be added vectorially to the
currents of the surrounding water masses on both sides in order to conserve a stable

Water Bodies and Stationary Current Conditions at Boundary Surfaces 459

equilibrium state, i.e. a uniform gradient current with the corresponding slopes of
the isobaric surfaces and of the sea level must be superimposed on the entire system
shown in Fig. 201. This will change somewhat the position of the isobaric surfaces and
that of the physical sea level. This circumstance should always be kept in mind in
dealing with the phenomena described in this section.

The oceanic structure in the boundary area between the Labrador Current and the
Gulf Stream to the south of the Newfoundland Banks is usually chosen as an example
for the oceanic structure presented in Fig. 201. Figure 202 shows a section through the

700

800

Fig. 202. Distribution of the specific volume anomalies in a meridional cross-section south
of the Great Banks of Newfoundland (according to Smith). Horizontal scale, 1:2 million;

vertical scale, 1 : 5000.

currents and the distribution of specific volume anomaly (Smith, 1926); the currents
here are approximately zonal ones (directed almost east-west). Disregarding the thin
top layer about 50 m thick, there is a heavier water body found in the middle flanked
to the north and south by water masses of greater specific volume. On the southern
side (Sts. 205 and 206) the lighter Gulf Stream water flows to the east (out of the plane
of the diagram), while on the northern side the water masses of the Labrador Current
flow towards the west in the area just to the south of the Newfoundland Banks
(Sts. 202 and 203). Figure 203 presents the topography, calculated from the mass
distribution, of some isobaric surfaces and of the physical sea level. As required by
theory, the presence of the heavier water body in the middle is shown by a low pressure
trough and at the surface by an elongated depression of the water level.

(b) Motionless Light Water Body Embedded into Moving Heavier Water Masses

The oceanic structure is also given here in the same way as for case (a) by the rules
for the stationary stratification of adjacent water bodies (Fig. 204). Here also the sea
level is lowest over the lighter water body, but this deep pressure trough diminishes

460

Water Bodies and Stationary Current Conditions at Boundary Surfaces

206 205 204 203 202 201

60 "^

" 40

I 20

Q 0-

Sea surface

50 dbar (m)

250 dbar (m)
450 dbar (m)

750dbar

Fig. 203. Distribution of the specific volume anomaly. Form of the physical sea surface and
of the isobaric surfaces in a meridional cross-section south of the Great Banks of New-
foundland.

in the deeper layers due to the wedge-shaped spreading of the adjacent heavier waters
underneath, which in the absence of the effect of Earth rotation would press upwards
the lighter water in the middle. The equilibrium of all the forces prevents this upward
movement and maintains the structure in a stationary state.

This simplest arrangement of water bodies is not readily found in ocean currents.
Dietrich (1935) in an investigation of the Agulhas Current found a mass distribution
which was similar to that pictured in Fig. 204, although with the current directions
exactly opposite that in Fig. 204. The pressure distribution as well as the topography
of the physical sea level would then be different. Dietrich assumed rising isobaric
surfaces towards the central lighter water body and no motion in the lighter body
(planar sea level and isobaric surfaces). The gradient currents in the adjacent heavier
water masses then correspond to the pressure field, but the current system as a whole
does not correspond to the rule of a stable position of the boundary surface. The
schematic representation given by Dietrich is in error.

^°^y ^^r

Fig. 204. Motionless lighter water mass embedded in moving hcaNier water (Northern

Hemisphere).

Water Bodies and Stationary Current Conditions at Boundary Surfaces 461

Figure 205 presents a dynamic section from Capetown towards the south-west based
on the "Meteor" observations (profile 1 a, 8-12 July 1925). The distribution of the
specific volume anomaly gives the structure shown schematically in Fig. 204. With this
stratification it can be expected theoretically (for the Southern Hemisphere) that there
will be a current flowing WNW to ESE just south of Africa (St. 20) and further

Fig. 205. Specific volume anomaly in a cross-section south-west of Capetown ("Meteor"
profile 8-12 July 1925, 34^ 49' S., 17° 48' E. to 4V 12' S., 11° 31' E.).

south (St. 18) there should be a current from ESE to WNW, if there is no motion
in the central region of the lighter water body. The observations show, however, that
this is not the case. According to dynamic calculations of the pressure field (Fig. 206)
there is a high-pressure ridge in the region of the central lighter water sloping down-
wards to the WNW in the northern part and to the ESE in the southern part. The system
of forces in the simple case of Fig. 204 is thus superseded by another pressure
system, which modifies conditions. It must be sufficiently strong to be able to reverse the
effect of the weaker opposite pressure gradient. These conditions can be represented in
a schematic way as shown in Fig. 207. Everywhere over the whole area the isobaric
surfaces and the physical sea level decline outwards though this is less so in the heavier
water masses than in the lighter central water. The current velocity in the heavier
water masses is thus less than in the lighter one in the middle. On the total northern side
there is a current from the east (the Agulhas Current), and on the entire southern side
is a current from the west (the West Wind Drift). The rule for a boundary surface
slope is now fulfilled ; since always when looking from the heavier towards the lighter
water the first moves towards the left relative to the latter (Southern Hemisphere).

Of particular interest is the application of the rule for the position of the boundary
surface between water bodies in subtropical and tropical seas, where the upper part
of the troposphere is to a large extent separated into two layers. The tropospheric
discontinuity layer separates an almost homogeneous top layer from the subtropo-
spheric water masses of only slightly diff'erent density. Here, in places, the transition

462 Water Bodies and Stationary Current Conditions at Boundary Surfaces

1000 dbor

Fig. 206. Position of the physical sea surface and of the isobaric surfaces in a cross-section
south-west of Capetown through the Agulhas Current and the West Wind Drift.

-. A

Fig. 207. Schematic representation of the oceanic structure in a cross-section normal to the
Agulhas Current and the West Wind Drift south of Africa. Shaded, heavier water masses;
non-shaded, lighter water masses; dashed lines, isobaric surfaces of this system; thin full
arrows in A and A', corresponding currents, in A from west towards east, in A' from east
towards west. Superimposed the pressure field of a water-"stau" in the central region; thin
full lines in the cross-section: isobaric surfaces, dashed in A, B and A', B': corresponding
currents, in A and B from east towards west, in A' and B' from west towards east. Thick full
lines in the section: resulting pressure field of the final current system. Thick full arrows
underneath: direction and speed of the resulting currents; in A and B from east towards
west, in A' and B' from west towards east.

Water Bodies and Stationary Current Conditions at Boundary Surfaces 463

layer carries the character of a real discontinuity surface and its position depends
principally on the currents in the top layer, since the cold water masses beneath are
almost motionless. Since the equatorial currents in both hemispheres flow from east
to west (North and South Equatorial Current) the dynamic equilibrium requires an
accumulation of the heavy water of the lower layer on the left side in the Northern
Hemisphere and on the right side in the Southern Hemisphere. The discontinuity
layer thus arches upwards in the equatorial regions and this must be associated with a
depression in the physical sea level at the equator. The vertical stratification of the
water bodies, the position of the isobaric surfaces and of the physical sea level is
presented schematically in Fig. 208a (Sverdrup, 1932, 1934a, Defant, 1936c,

{a)S

Equator

Fig. 208. Different positions of the thermocline and of the physical sea surface in the

tropics and subtropics and the corresponding current systems (according to Sverdrup).

IV, current towards west; E, current towards east.

p. 315). When the currents are symmetrical about the equator, this stratification will
also be symmetrical. However, neither of these conditions actually occur in the
Atlantic nor in the Pacific and very probably also not in the Indian Ocean. The thermal
equator is at times found north of the geographical equator so that the equatorial
currents are not symmetrical about the equator. In the Indian Ocean the thermal
equator lies to the south of the equator during the southern summer. This complicates
the adjustments of the boundary surfaces, since the Coriolis force, the effect of which
is symmetrical about the equator, acts as a counter force to the non-symmetrical
pressure field. An accumulation of the subtropical water masses, asymmetric to the
equator, more or less as in case b in Fig. 208 with the position of the physical sea
level and of the isobaric surfaces indicated there, cannot be stable. This is because, for
stable stationary conditions, the topography of the sea level and of the isobaric surfaces
at the equator must always show either a maximum or a minimum. In case b there will
be a current from the west on the southern side of the equator and a current towards
the east on its northern side, and at the equator itself the velocities will be infinite

464 Water Bodies and Stationary Current Conditions at Boundary Surfaces

(disregarding friction). The current system can only be stabilized by distributions
pictured in cases c and d. In both cases a counter current flowing eastward must be
introduced between the westward flowing equatorial currents of the Northern and
Southern Hemisphere. In case c it lies entirely within the Northern Hemisphere,
together with parts of the South Equatorial Current which extends across the equator;
in case d the counter current is broader and extends somewhat across the equator into
the Southern Hemisphere. This kind of adjustment position of the pressure surfaces
and of the boundary surface, thus satisfies the requirements of a boundary surface
slope for moving water bodies.

These theoretical considerations can be tested by using the available observational
data. Figure 209 presents for a meridional profile, along the strongest inclination of the
surfaces, the topography of the pressure surfaces and of the physical sea level of the

*20
dyncm

♦ 70 -

0 —

-10 -

-20 —

Z(f S 10°

10° N 20°

./•

V-7-

-^130
160

■i200

J i

Fig. 209. Meridional cross-section through the Atlantic Ocean (25° N. to 25° S., 20° W. to

30° W.). Upper picture: physical sea surface (relative to the lower current 400:1 exaggerated).

Lower picture: depth of the thermocline (tropospheric discontinuity layer).

Atlantic Ocean. The structure shown corresponds entirely to that given in Fig. 207c.
There is no doubt that the adjustment of the tropospheric discontinuity layer is
dynamically controlled and to a large extent imposed by the arrangement of the ocean
currents in the top layer ; there seems to exist a very close mutual adjustment between
them. The observed slope agrees not only quantitatively but also qualitatively with
that required by theory (Table 132). If the slope of the boundary surface, given in
metres per 3 degrees of latitude, is denoted by / and that of the physical sea level
by /i, then taking a mean value for pi of 1-024 and putting /= / x 10^^ the formula
(XIV.8) gives

3-46

and

/. =

-9-77 X 10-* m.

The value for r^ is taken as the approximate average over the entire top layer. The
observed and calculated values are nearly equal.

Water Bodies and Stationary Current Conditions at Boundary Surfaces 465

Table 132. Slope of the tropospheric transition layer and the physical sea
level in the North and South Equatorial Current in the Atlantic Ocean.

5^S.

17-5° N.

24 0

24-6

CTo

26-5

a., — CTi

2-5

1-9

Vi (cm/sec)

6-5

Theoretical value: m=26-4m; mi=— 2-58cm; m — 52Q m; m^
Observed value: A?2 = 26-4m; Wi=— 2-4cm; A?7 = 52-5m; nii

— 51 cm
-50 cm

When the water masses in a two-layered ocean are in rotation they will be subject
to a centrifugal force in addition to the gradient and Coriolis forces. Under stationary
conditions these three forces must balance. Such systems of rotating water masses
have been examined in detail by Exner (1917) and especially by Bjerknes (1921).
When the motion is symmetrical around the rotation axis, the vortices are termed
"circular vortices". In cylindrical co-ordinates r is the distance at right angles from the
axis of rotation z (positive downwards) and c is the rotational velocity (at right angles
to r, positive for cyclonic and negative for anticyclonic motion). For a non-accelerated
current (c = 0) the following quantities can be introduced in the boundary surface
equation (XIV. 5):

X^=fc^+ j;

Zl=g; A'2=/C2 +

Zz = g.

c'^lr is the centrifugal force, which must be taken into account for curved trajectories.
The slopes of the pressure surfaces, of the physical sea level for the lighter and the
heavier water and of the boundary surface can be determined, and it is obtained

tan /Si

tan ^2

f c^

■L r — -^

'-2

g rg

(XVI.9)

and

tan y = —

/ P2^
g Pi

PlCl

P2C2

The third equation can be somewhat simplified
Ac = C2 — Ci

Pi Ac

P2 — Pi

With sufficient accuracy, when

/
tan y =

P2 — Pi

(• ~ ^)'

(XVI. 10)

On comparison with formula (XIV.8) it can be seen that the effect of the centrifugal
force is contained in the expression in brackets. The difference between the slope of
the boundary surface in a rotating flow from that in a straight current remains small;
assuming /= 1 X 10'* (about 45° latitude), r = 100 km and Ci + Cg = 40 cm/sec,
then the expression in brackets gives 1-04, that is, an increase of about 5% can scarcely

466 Water Bodies and Stationary Current Conditions at Boundary Surfaces

be expected for extensive vortices. For small vortex sizes, however, it may be as large
as 30-40%. A difference from the case for straight flow exists in so far as the slope of
the boundary surface depends on the distance from the axis of rotation. In general,
when the area immediately around the axis of rotation is disregarded, the oceanic
structure of a circular vortex of this type can be readily derived from the rule given
above. Four cases can be distinguished (Fig. 210, Northern Hemisphere).

(o)

(b)

Fig. 210. Rotational symmetric stationary vortex in a two-layered ocean (position of the
boundary surface and form of the isobaric surfaces, physical sea surface, respectively).
a and c, cyclonic and anticyclonic rotation in case of a faster rotation of the upper layer.
b and d, cyclonic and anticyclonic rotation in case of a faster rotation of the lower layer
(underneath the sections diagram of forces for only one point of the lighter and heavier
water mass. G, gradient force; C, Coriolis force; Z, centrifugal force).

Case a: Ac < 0, for cyclonic rotation Cg < Ci: tan y > 0. The boundary surface
rises towards the centre, in fact more rapidly near the vortex axis and less further out;
tan ^, on the other hand, is negative in both layers, that is, the pressure surfaces and
the physical sea level rise outwards, more so in the upper than in the lower layer.
This is the case of a cyclonic vortex with the upper layer rotating more rapidly. Due
to the rotational effect the heavier water accumulates around the axis of rotation while
the lighter top layer is forced to the outside. In the central area there is a depression
in the physical sea level and the isobaric surfaces.

Case b: Ac > 0, for a cyclonic rotation Cg > Ci: tan y < 0. tan /S is negative in both
layers and the boundary surface, the pressure surfaces and the physical sea level rise
towards the outside; cyclonic vortex with the lower layer rotating more rapidly. The
lighter water masses accumulate around the vortex axis and there, as in the previous
case, the physical sea level and the pressure surfaces show a depession. In these cyclonic
cases the sum of Coriolis force and the centrifugal force act towards the outside and
a larger gradient force is required to balance this combined action. The boundary
surface slope must therefore be greater than for water bodies arranged in strips.

Water Bodies and Stationary Current Conditions at Boundary Surfaces 467

Case c: Ac > 0, for anticyclonic rotation |ci| > [ca]. As long as the term in brackets
in (XIV. 10) remains positive, which is always true except in extreme cases, then
tan y < 0 and the boundary surface rises towards the outside. Tan ^ is positive in
both layers and the slope of the pressure surfaces is less in the heavier water body than
in the lighter: anticyclonic vortex with the top layer rotating more rapidly and a central
dome-like uplift of the pressure surfaces and of the physical sea level. The rotation
gives rise to an accumulation of the lighter water masses around the rotational axis.

Case d: Finally, it is possible in an anticyclonic rotation to have /Ic < 0 and then
kal > kil- The slope of the boundary surface rises towards the centre since tany is
positive (with the same restriction as in case c). The pressure surfaces also rise towards
the centre but in this case more strongly in the heavier than in the lighter water layer :
anticyclonic vortex with the lower layer rotating more rapidly and a central dome-
like uplift of the sea level and the isobaric surfaces. Here the lower heavier water
accumulates around the vortex axis. Since in the sea the current velocity almost always
decreases with depth, cases a and c will predominate. In a cyclonic vortex the deep
water is hfted close to the surface and if the vertical velocity gradient is sufficiently
large the boundary layer may reach the surface. Then the vortex centre will be filled
with deep water. In an anticyclonic vortex, on the other hand, there is an accumulation
of the hghter upper water around the vortex axis that may extend downwards to con-
siderable depth.

The actual stratification in the sea seldom consists of only two layers; the same laws
apply, however, also to a continuously stratified ocean (see Chap. XV). The boundary
surface slope is then replaced by the slope of the isosteres and in place of sharp kinks
there appears a steady curvature in the isobars. Also here, due to the low velocities
and the large radia of curvature of the current trajectories, the centrifugal force is of
little importance compared with the Coriolis force for an estimate of the mass field
adjustment. Figure 21 1 shows dynamic sections through such cyclonic and anticyclonic

Fig. 211. Mass and pressure distribution in rotationally symmetric layered vortices with a
decreasing rotational velocity with depth, {a) Cyclonic; {b) anticyclonic rotation.

circular vortices in a stratified ocean; in both cases it is assumed that the velocity of the
current decreases with depth; for a two-layered ocean they correspond to the cases
a and c of Fig. 210.

Charts of ocean currents often show more or less extensive vortices in the top
layers. They are found mostly in those areas where the wind field also indicates

468 Water Bodies and Stationary Current Conditions at Boundary Surfaces

rotational (cyclonic or anticyclonic) motion. The anticyclonic winds around the sub-
tropical high-pressure centres thus give rise in both hemispheres to anticyclonic large-
scale vortices between the oceanic West Wind Drift and the Equatorial Currents.
These are elongated corresponding to the shape of the high-pressure cells and take the
form of a broad convergence zone. In the central parts of these anticyclonic vortices
there is always a mass distribution corresponding to that in Fig. 211 b; that is, with
an accumulation of lighter water in the central part of the convergence area. Condi-
tions of this type are particularly well developed in the North Atlantic, where there is
an accumulation of warmer water with a corresponding depression of the isosteres to
600-800 m ; the isobaric surfaces and the physical sea level show a corresponding
uphft.

Large-scale vortices with cyclonic sense of rotation are found in the intermediate
region between the oceanic West Wind Drifts and the Polar Currents; that in the
North Atlantic between the Polar and the Atlantic Current. Here the actual oceanic
structure will be very nearly that pictured in Fig. 2\l a, which shows that the isosteres
arch upwards. Such cases will be referred to again when discussing the current con-
ditions in particular oceanic regions,

A very typical case of a smaller-size cyclonic vortex was observed in the Gulf
Stream just north of the Azores above the "Altair" cone during the International
Gulf Stream Survey, 1938 (Defant, 1940 b). The centre of the vortex was found in
upper layers a little south of the greatest submarine elevation; in deeper layers it
appeared directly above the cone. All the vertical oceanographic sections show this
vortical disturbance and its vertical structure. Figure 212 presents a somewhat smoothed

100

200

300

E 400

500

600

700

800

900

Fig. 212. Meridional density section through the cyclonic vortex above the "Altair'
submarine volcano in the Atlantic Ocean (somewhat smoothed).

Water Bodies and Stationary Current Conditions at Boundary Surfaces 469

density section. Although the axis of the vortex is somewhat inclined towards south,
current measurements and the mass distribution suggest a subdivision of the total
vortex into two parts or systems.

(1) The upper system down to 100-150 m depth includes a discontinuity layer at
about 25 m. The velocity of the basic current in the top layer is about 15 cm/sec and
in the denser lower layer, however, 20 cm/sec. This is thus a strongly stratified cyclonic
vortex with a speed of rotation increasing with depth. Under steady conditions the
isosteres must therefore dip downwards in the lighter water masses which are con-
centrated around the vortex axis. This is shown very clearly by the section given in
Fig. 212.

(2) The lower system extends through the layers below 150 m, where there is a normal
increase in density with depth and a steady decrease in the velocity from about
20 cm/sec at 200 m to about 6 cm/sec at 800 m. This is therefore a weakly stratified
cyclonic vortex with decreasing rotational velocity with depth. The required uplift of
the isosteres (accumulation of lower denser water around the vortex axis) is again
obvious from Fig. 212.

Also quantitatively the observed slopes are in a good agreement with that required
by theory (equation XIV. 10). Since <^ = 44° 33' N.andtherefore/= 1-023 x lO"* sec-^
equation (XIV. 10) gives for the upper system: a^ = 26-30, q = 15 cm/sec, ag = 26-65,
Cg = 25 cm/sec ; the isosteres slope downwards towards the centre by 92 m in 60 km ;
observed 70-90 m. For the lower system: ct^ = 26-8, Ci = 20 cm/sec, cto = 27.5,
Cg = 6 cm/sec; the isosteres slope upwards towards the centre by 214 m in 100 km;
observed 230-290 m.

The cyclonic vortex performed pulsations, as was indicated by the observations made
at the anchor stations. The period of these pulsations corresponded to the period of
inertia oscillations (see p. 472).

Sandstrom (1914, 1918), has carried out laboratory experiments to test the effects
of cyclonic and anticyclonic air currents on stratified water masses underneath.
Reference is made to these in this connection.

4. Up- and Down-gliding Surfaces: Pulsations of Stationary Vortices

In systems of moving water bodies for a stationary position of the boundary
surfaces there will be no vertical motions according to equation (XIV. 7). If the equili-
brium conditions are not satisfied, accelerations will occur and as a consequence
vertical motions are generated which will lead to changes in the position of the dis-
continuity surfaces. If the slope angle of the boundary surface is denoted by e and
differs from that for its stationary equilibrium y, then e will tend towards its equili-
brium slope y. If the boundary surface is steeper inclined than in the equilibrium
state (e > y), in order to reduce e the upper lighter water must spread out over the
lower heavier water and the lower one will intrude underneath the lighter. Above the
boundary surface there will be an up-gliding and below it a down-gliding (up-gliding
surface).

If, on the other hand, for e < y the reverse will apply. In the lighter water type there
will be down-gliding and in the heavier up-gliding (down-gliding surface). The processes
occurring at the boundary surface can be decisively influenced by the initiated vertical
motions. Exner (1924) and J. Bjerknes (1924) have investigated the processes that may

470 Water Bodies and Stationary Current Conditions at Boundary Surfaces

occur at arbitrarily inclined discontinuity surfaces. Taking horizontal accelerations
into account but neglecting the very small vertical accelerations (w^ = W2 = 0) and if
the boundary surface is parallel to the >'-axis having an inclination tan y then equation
(XIV.5) gives the relations (^-positive upwards) :

(Pi^i — P2W2) =f{pii\ — P2V2) — Sipi — P2) tan € and Pii\ — p.iV2 =f(piUi — p^Uo)

(XIV. 11)

Near the boundary surface the velocity in each of the water bodies will be tangential
to it : Wi = Ml tan e and vt-g = Mo tan e, so that

PiH'i — P2H'2 = (pith — P2W2) tan e.

(XIV. 12)

These equations form the basis of the dynamics of up- and down-ghding surfaces.
If in the first of these equations e = y (stationary boundary surface condition), then
P2W2 — P2W2 = 0 and from (XIV. 12) it follows that p^Wi = P2^2- On the other hand,
according to the second part of the equation (XIV. 1 1)

PlVl — P2«2 ^0.

This implies that : tip- and down-gliding can also occur at stationary boundary surfaces
if the currents are accelerated also in the direction parallel to the gliding plane. If the
mutual adjustment between current velocities and stable position of the boundary
layer gets disturbed by changes in the velocities, then up- and down-gliding motions
must occur along the boundary surface in order to preserve a stationary state of its
inclination. Thus when

(0 Pi'"i — />2i'2 < 0: piMi — P2W2 > 0 and p^w\ — P2**'2 > 0

and when

(2) pjt'i — /Da^a < 0: piu^ — p^Uo < 0 and p^w^ — p<^<2, < 0.

In the first case where there is a stronger acceleration in the lower water mass along
the positive j'-axis than in the upper, an up-gliding surface is to be expected. In the
second case, however, where there is a stronger relative acceleration along the positive
j-axis in the upper water mass, there will be a down-gliding surface. These two cases
are illustrated in Fig. 213; they apply for the Northern Hemisphere. In the Southern

Fig. 213. Stationary up-gliding (to the left) and down-gliding surfaces (to the right)

(Northern Hemisphere).

Water Bodies and Stationary Current Conditions at Boundary Surfaces 471

Hemisphere the arrow-directions indicating the velocities and the accelerations parallel
to the boundary surface have to be reversed.

So far the discussion applies only for infinitely extended boundary surfaces. If they
intersect the sea surface (fronts) or the sea bottom the up- and down-gliding motions
will give rise to horizontal water currents in its vicinity and consequently to changes
in the position of the boundary surface.

Cases of this type can be found at the oceanic polar fronts. Figure 2 14 shows the polar
front between the East Greenland Current and the Atlantic water to the south of the
Denmark Strait. The mass distribution requires larger velocities in the polar current
towards the south and smaller ones in the Atlantic water as is found by observation.

Polar front

(a) s.

Fig. 214. Oceanic vertical stratification and currents at the East Greenland oceanic polar
front. Picture to the left: up-gliding of the polar water and down-gliding of the Atlantic
water for an accelerated East Greenland Current : boundary surface progresses towards east.
Picture to the right: down-gliding of the polar water and up-gliding of the Atlantic water for
an accelerated Atlantic current: boundary surface progresses towards west.

In general, there exists a stable equilibrium in the current system between the mass
structure and the currents with a stable boundary surface position. If, however, an
easterly wind piles up polar water ("Anstau") along the east coast of Greenland, or if
other conditions in the North Polar Sea cause an increase in the strength of the East
Greenland Current, then the water masses of the current will be accelerated towards
the south and the boundary surface will become an up-gliding surface (Fig. 214 a).
This up-gliding along the boundary surface in the lighter polar water mass must come
to an end at the sea surface ; here it gives rise to a reduction in the inclination of the
boundary surface, that is, the extent of the East Greenland Current at the surface will
increase and will force the Atlantic water masses seaward.

In the opposite case (Fig. 214 6) if the Atlantic water is accelerated towards the
north, the boundary surface becomes a down-gliding surface. It thus becomes steeper
and the extension of Atlantic water is increased. Pulsations in the basic currents will
be associated with variations in the mass distribution. The large-scale aperiodic
atmospheric disturbances of these regions must be accompanied by corresponding
large changes in the oceanic structure and the ideas outlined above are of major
importance in the coupling of these two phenomena.

Similar conditions must apply for the much longer polar front in the Southern
Hemisphere. Here the temperature is the decisive factor for the mass structure and
the boundary surface between the West Wind Drift and the South Polar Current
slopes downward towards the north (towards the equator). In order to secure stationary

472 Water Bodies and Stationary Current Conditions at Boundary Surfaces

conditions, the West Wind Drift must have a greater velocity towards the east than the
South Polar Current to the south of it, which is also directed east. Since here also
disturbed meteorological conditions are frequent in this region, the varying influence
of the action of the atmospheric flow will sometimes accelerate the oceanic West
Wind Drift and sometimes the South Polar Current, and therefore the polar boundary
surface will change from an up-gliding to a down-gliding surface and back again and
there will be corresponding displacements of the polar front in meridional direction.
These processes seem to continue nearly all the time and may be associated with the
observed sinking process of large water quanta of sub-Antarctic waters. This process
is most probably of a pulsatory character and is definitely the source of the sub-
Antarctic intermediate water penetrating far to the north.

Variations of the boundary surface can also arise in circular vortices if there are
changes in the vertical current structure. If (see in Fig. 215) for example, the boundary

Fig. 215. Pulsations of a circular vortex in cyclonic rotation.

surface and the physical sea level lie in the position 1-1 under average conditions, then,
if the velocity between the upper and lower water bodies increases, there will be greater
accumulation of the lower water type around the axis of the vortex and the inclination
of the boundary surface will increase (position 2-2). If, on the other hand, this diff"er-
ence becomes less, then the accumulation of lower water will be dispersed and the
inclination will decrease. Periodic variations in the mass structure will thus occur in
the vortex; the boundary surface and the physical sea level will oscillate ^round a
mean position and these oscillations will have the character of standing waves (see
Vol. II).

In the cyclonic vortex over the "Altair" submarine cone in the Gulf Stream north
of the Azores (see p. 454) periodic variations of this type were present both in the oceanic
structure and in the vertical current distribution. They were very well developed in the
upper system and of a period corresponding to the inertia period (17 n). Since the
periodic variations in the current amounted to as much as half of the velocity of the

Water Bodies and Stationary Current Conditions at Boundary Surfaces 473

basic current, the changes in time of the distribution of the isosteres must have been
quite considerable. Figure 216 shows these changes in the vertical current structure in
the two layers of the upper system: 5-15 m and 30-100 m. In the lower part of the
vortex the velocity is greatest between 2 and 3 h and at the same time least in the
upper part. During this time-interval there is thus an increase with depth of the velocity
of rotation. In the interval between 9 and 16 h conditions are reversed; at 10 h the

8 16 24 32

cm/sec

Fig. 216. Changes in the vertical structure of the current of the upper system in the cyclonic

vortex above the "Altair" submarine volcano. •« — , current in the layer between 5 and 15 m

depth; <=, current in the layer between 30 and 300 m depth.

top layer has the greatest velocity and there is thus at this time a decrease in the
rotational velocity with depth. This feed-back of these oscillations of the current field
on the mass distribution in the vortex must be extremely strong to give a complete
reversal of the current structure. At 10 h there must be an increase of the up-lift of
the isosteres and at 2 h an increased depression. These oscillations of the isosteric
surface about nodal lines at a certain distance from the vortex centre have been
demonstrated by observations of the anchor station. The isotherms and isohalines
oscillate around a mean position with an inertia period of 17 h, so that the anchor
station must be somewhat displaced towards the outer edge of the vortex, because the
isosteres are always lowered at 10-5 h and always lifted at 2 h.

The oscillations in a circular two-layered vortex can be accounted for theoretically
(Defant, 1940 b) and an estimate can be made of the period of the free oscillations of
such a system. If the effect of centrifugal force is neglected (it is always small) then the
mean position of the boundary surface in such a vortex will correspond to the follow-
ing relation (z positive upwards; centre of the vortex at .v = 0; horizontal extent of
the vortex = 21):

474 Water Bodies and Stationary Current Conditions at Boundary Surfaces

z = /?2 + S cos (77// )x, (XIV. 1 3)

5 f (p^ih— Pi«i)

where o —

g Pi — Pi

(Margules boundary surface slope).

If small periodic variations (disturbance values) are imposed on this equilibrium
system in the currents Ui and u.^, then the boundary surface will oscillate about
its steady-state position. As a consequence in the most simple case these oscillations
will give rise to upward and downward movements in the central part of the vortex
with a phase exactly opposite to that of the outer vortex portions. To the equation
(XIV. 13) will thus be added an additional periodic term of the form

Z — A COS -J- COS a J, (XIV.14)

whereby C7„ is the frequency of the free ("Eigen") oscillation (period T = Irrjan',
n = 1, 2, 3, ... , gives the number of node-points in the oscillating system).

When corresponding boundary conditions are taken into account the equations of
motion give an equation for the determination of the frequency a„ of the "Eigen"
oscillations of the oscillating vortex as a function of the dimensions of the system.
The following equation is obtained

where //^ and Ju are the thicknesses of the two layers and 2/ is the total horizontal
extent of the vortex. These "Eigen" frequencies depend in a characteristic way on the
angular velocity of the Earth. If the Earth were not rotating (/= 0) then the period
of the free oscillation would be given by

277 _ 2/ ///

cTr ~ n \]\

In 11 llpilh + Pilh

g(p2 - Pi)

(XIV. 16)

This is a period for an internal standing wave in a two-layered water mass of an
extent / (see Vol. II).

If for large dimensions of the oscillating system the period Tr for a non-rotating
Earth is large, then the second term in the equation(XIV.l 5) will be so small as compared
with/2 tjj^t jt can be neglected and the longest "Eigen" period of the system will be
equal to the inertia period.

Ti = ha pendulum day = — ^ . (XIV. 17)

If the second expression accompanying/^ in the equation (XIV. 1 5) cannot be neglected,

when (r^ > Ti),

when (Tr < T,).

then it is obtained with sufficient accuracy

T= Ti

[' ^ ' (SI

however,

T=Tr

[' - m

Water Bodies and Stationary Current Conditions at Boundary Surfaces 475

In most cases in the ocean T^ <^ Ti, so that the ''Eigen" period of such an oscillating
system will always be close to the inertia period. In the vortex over the "Altair" cone
2/ = 120 km: the mean densities of the upper and lower layer pi and po are 1-0263
and 1-0283 and/- 1-023 x 10"* sec-\ then for hi = 30m and h^ = 1000 m it is
found that T, = 17-1 h and the "Eigen" period of the system according to equation
(XIV. 1 5) is r = 1 6-76 h. Thus the "Eigen" period of the vortex over the "Altair" cone
approaches closely the period of an inertia oscillation, as was found by observation;
inertia oscillations are merely the free oscillations of an enclosed sea the equihbrium
state of which has been disturbed. They are probably set up by external causes especially
by meteorological conditions (hke storms and similar phenomena). In this particular
case a storm occurring just before the anchoring of the "Altair" seems to be the cause
for the pulsation of the otherwise stationary vortex above the "Altair" submarine
volcano.

Chapter XV

Ocean Currents in a Non-homogeneous

Ocean

1. Introduction

If all the external forces that may act on the sea are excluded, ocean currents can still
be produced by internal forces. Differences in the mass structure will represent an
internal system of forces that will act until the resultant mass displacements lead to the
establishment of a mass distribution corresponding to that of a static equilibrium.
It is customary to denote ocean currents generated by such internal forces as "con-
vection currents" although they have nothing to do with oceanic convection pheno-
mena. In order to avoid this unsuitable notation it seems to be advisable to call them
"density currents", since they depend solely on the three-dimensional difference in
the density field. Treatment of these density currents involves greater difficulties than
that of drift and gradient currents, in particular, since the external forces (wind and
atmospheric pressure) can be regarded as independent from the currents themselves,
while the density currents and the density differences producing them influence each
other. Furthermore, the density anomalies, being internal forces, are distributed three-
dimensionally in space, while wind and atmospheric pressure at the sea surface act
only in two dimensions.

The beginnings of a theory of density currents goes back to Mohn (1885, 1887)
whose work can without doubt be described as "the beginning of a new era in physical
oceanography" (Helland-Hansen and Nansen, 1909, Vol. II. 2, p. 390). However,
this theory, the aim of which was rather wide-spanned, was incapable of influencing
the further development of theoretical oceanography, since it was running far ahead of
the development of oceanography, which at that time made its progress mainly along
geographical lines and because the defects in it were difficult to eliminate. It was soon
forgotten (Thorade, 1925). The foundation for a firmly founded theory of density
currents was provided by the application of the Bjerknes theorems of vortex formation
and circulation acceleration to oceanographic problems. Thereby it was necessary to
leave aside classical hydrodynamics, dealing only with homogeneous media, and to
make use of physical hydrodynamics where the media had a full physical reality.
Some of the results were later derived directly from the hydrodynamic equations of
motion. These derivations are, in part, clearer and more comprehensible, and it
therefore seems advisable to discuss the simpler problems first.

2. Relationships Between Current and Density Fields in a Horizontal plane. The law

of Parallel Fields

A general relationship between density and current fields can be derived quite
simply (Defant, 1931). In general, the vertical component of the velocity, that is,

476

Ocean Currents in a Non-homogeneous Ocean All

the vertical slope of the stream lines is so small that the current field can be regarded
as horizontal. Under stationary conditions the stream lines follow the stream function
i/rCxj') = Ci; the horizontal density distribution shall be given by p{x,}') = c^. The
angle between the two sets of curves may be y. If the stream lines are at an angle a to
the positive .Y-axis and correspondingly the isopycnals at an angle /3, then

difj Idip Sp /dp

tan a = — K-l^^ and tan j8 = — -- /
c.v/ dy dxj

From this it follows that

dip dp dip dp

^ oj^^Z_^y^^ (xv.i)

dijj dp dill dp
dx dx dy dy

If the stream hnes are parallel to the density lines (y = 0), then consequently

dj^d_P_djPd_p^^ (XV 2)

dx dy dy dx

Disregarding for the moment the effects of friction (turbulence), and if there are no
physical changes in the water masses due to external circumstances then, for stationary
conditions dujdt = dv/dt = 0, the equations of motion (XIII. 1) will also apply for a
non-homogeneous sea. Eliminating the pressure p and taking into account the con-
tinuity equation and introducing a stream function (equation X.35), equation (XV.2)
is obtained. In a non-homogeneous sea stationary conditions require that the stream
lines and the isopycnals (isosteres) are parallel. This result is self-evident since otherwise
these surfaces would be displaced and this would contradict the condition of a
stationary state. The same also applies to isothermal and isohaline surfaces. On the
other hand, the following equation can be derived from the equation of motion and
the hydrostatic equation (Ertel, 1933)

/ dpu dpv\ d^p dp d^p dp

^' Y' -dl ~ P""^) = ~ W^ dx-^ ^^z dy-

By means of the hydrostatic equation

-dz=^^P

this equation can also be written in the form

■' P dz \vj ^ \dx dy dy dx]

If the total velocity V is at an angle x to the ^--axis so that u = V sin x and v = V cos x
then

If the isobars and isopycnals are parallel in a horizontal plane, then the expression in
brackets, D, is zero. The mass field is therefore barotropic and dxjdz = 0, that is, the

478 Ocean Currents in a Non-homogeneous Ocean

current does not turn with depth, or the current directions at all depths will lie in
one and the same vertical plane. Since for frictionless motion the current follows the
isobars and these coincide with the stream lines, D will be identical with equation
pCV.2). Except at special disturbance locations (discontinuity surfaces, discontinuity
layers and fronts) the stream Hnes therefore will also coincide with the isolines at all
depths.

If turbulent friction should also be taken into account, it is necessary to go back to
the general equations of motion and elimination of p leads to the equation

P^e.^a^s,^,^ (XV.4)

dx cy By ex j oz^

For a simple potential flowzJ 0 = 0 and the condition of parallehsm of stream lines and
density lines still applies. If, however, a vortical motion has to be dealt with, this
parallelism will be lost.

The angle at which they intersect will depend on the turbulence and on the water depth. It can be
shown that now

tan y = -^r— ,

where I, = dvjdx — duldy denotes the vertical vorticity component. If the co-ordinate system is placed
in the direction of the average current, then f = 0. At the sea surface assuming a linear pressure
gradient (Ap = 0) and a decrease of velocity with depth u = \a z^ (sea bottom z = 0) as well as a
depth of water h, is obtained

tan y = j— .
fpir

For fflp = 200 cnr/sec and/= 10-^ sec-i (at about 45° N.)

tany=(^)

if the depth of water H is measured in metres. For a large water depth y will be almost zero; if the
water is shallow (shelf seas) it may reach values of 10-20°.

Summarizing, it may be stated that for steady frictionless currents in a non-homogen-
eous sea the isolines of the different oceanographic factors and the stream lines must
coincide, but in the presence of strong turbulence especially in shallow seas this
parallelism is lost.

Attempts have very often been made in oceanography to deduce the current field
from the distribution of the temperature and the salinity and other factors. In general,
such deductions are permissible and the method gives results corresponding reasonably
with reality, but deductions from isoline charts should not be taken as more than
indications of the rough course of the currents. However, exactly at the point where the
current field is of particular interest (near discontinuity surfaces and fronts) the method
fails completely (Castens, 1931).

These arguments are connected with the "law of parallel fields" (Helland-H.\nsen
and Ekman, Ekman, 1923). Comparison of the distribution of the oceanographic
factors at different depths shows the striking phenomenon that the isolines at any
particular depth are parallel to each other, and moreover that they are parallel also

Ocean Currents in a Non-homogeneous Ocean 479

with those in deeper layers. This agreement in the course of these lines also extends
to the dynamic isobaths at any depth. It must therefore be concluded that the current
vectors are also tangential to all these sets of curves and that there is complete equahty
between all these hnes. This law allows deduction according to the Ekman theory of
the direction of the deep current outside the upper and lower frictional depth which
represents the layers in which the drift current and the bottom current are found. All
modem cartographic representations of the horizontal distribution of these factors at
different depths confirm the general validity of this law (see, for example, the ''Meteor'''
Reports, Vol. VI, Atlas).

The basic prerequisites for the vahdity of this law are the same as in the rules derived
above for the relationships between the oceanographic factors and the current field
in any horizontal plane. These are satisfied for the deep currents except in those areas
where they are disturbed by discontinuity layers, or where due to mixing processes
there caimot be any stationary spatial density distribution.

3. Horizontal Steady Currents in a Stratified Ocean

The dependence of the vertical velocity distribution in a current on the stratification
of the water masses in the pressure field is already shown by the behaviour of two
adjacent water bodies. In steady state continuous changes in density require also a
definite mutual adjustment between the mass and pressure field. If the flow is directed
along the positive >'-axis, then for a steady frictionless motion

Inserting the hydrostatic equation g = a(8pldz) (z counted positive downwards),
elimination of p leads to the relation

8v 8 log a ? 2 log a

p- = ^ — p^- - 7- ^^ • (XV.5)

8z 8z f 8x

This states that for a given vertical and horizontal mass distribution there will always
be a vertical velocity distribution given by (XV.5). Introducing the slope of the isobaric
surfaces tan ^ = — {flg)v and that of the isosteric surfaces tan y = — {8pl8x)l(8pl8z)
the equation takes the form

dv 2 8 log a

^ = -^(tan y - tan j8) -^ . (XV.6)

8z J cz

Since 8 log aj8z is always negative, the expression in parenthesis decides about
increase or decrease in the velocity with depth. In other words, this increase or decrease
in velocity depends on the difference in the slope of the two intersecting sets of surfaces
or lines in a dynamic section. Figure 136c (page 331) shows the two possible cases (r
is always positive); in that shown on the left-hand side the expression in brackets
is always positive, and therefore 8vl8z < 0, or there will be a decrease in velocity with
depth. In the case on the right-hand side 8vj8z > 0, and there will be an increase in
velocity with depth. When y = ^ then 8vjcz = 0 which is the barotropic case with a
constant velocity at all depths. These results can be expressed by the following rule :

480

Ocean Currents in a Non-homogeneous Ocean

If the isosteres slope downwards {upwards) from left to right when facing downstream,
then a steady current will show a decrease {increase) in velocity with depth {Northern
Hemisphere).

It can be seen that equation (XV.6) allows a determination only of the vertical
velocity differences and it does not give the velocity itself and thus affords only relative
velocity difference distributions in vertical direction. This state of affairs recurs in all
similar cases and is a consequence of the indeterminate nature of the problem.
Equation (XV.6) has been derived from the equations of motion alone; to determine
the entire state of motion completely requires the continuity equation. Only then are
the conditions uniquely defined.

Equation (XV.5) can be written also in another form:

oz cz

gda

fdx

This can be used for a step-wise calculation of the vertical velocity distribution from
layer to layer (Defant, 1929 b).

If at two stations separated by a distance L at a depth r = 0 the specific volumes are
tto and a'o and at a depth z = h a-^ and a'^, the following formula can be used for a
numerical determination of the velocity difference ^o ~ ^i

(ai + a'i)ro — (tto + a'o)fi = j^ {a^

a'o + «i — a'l)-

(XV.7)

Table 133 contains the specific volumes at six depths down to 750 m for the stations
205 and 206 on the section through the Gulf Stream and the Labrador Current south
of the Newfoundland Banks (Fig. 202). For 0 = 40° 10' and L = 59 km the equation
(XV.7) gives the vertical velocity on the assumption of no motion at a depth of 750 m.

Table 135. Calculation of the vertical velocity in the Gulf Stream south
of the Newfoundland Banks

Depth

St. 205

St. 206

L= 59kni

(ag — a'o) + (aj — a'l)

(m)

a — a

(m)

(cm'sec)

0-97393

0-97449

56(xlO-^)

107(>;10-«)

64-7

363

414

102

59-7

125

312

363

125

112

52-5

250

217

278

200

101

39-3

450

119

159

300

20-3

750

0-96973

005

Werenskjold (1935, 1937) has developed a simple and practical method for the
same objective. Neglecting in equation (XV.6) tan ^ in comparison with tan y = /,

Ocean Currents in a Non-homogeneous Ocean

481

which is always permissible and integrating it between level ro(po.io) ^iid the level
^liPx.v^ gives

i\ =

idp.

(XV.8)

whereby p,„ is a mean density for the layer r^ — Zq. Denoting the tangents of the slope
angles of the isopycnals or isosteres drawn in a dynamic section with intervals A p
and Aa, respectively, by /, then equation (XV.8) can be transformed into the simple
relation

S A p _ _ s Aa

t\ =

nf p,

nf a,

ZJ.

(XV.9)

The summation has to be taken over all the isopycnals or isosteres which cut a given
vertical line between levels Tq and z^ and n is the vertical exaggeration of the section.
Values of 7 can be read directly from the section using a transparent scale (Fig. 217).
If the isopycnals in a vertical section are plotted at intervals of 10~* and the isosteres at
intervals of 5 X 10""^ and if the vertical scale of the section is 1 :2500 and the horizontal

\AV

;♦

v\v

VA\

\\v

\\W

\\\

\V^

\\\

\\W

VA\

\\V

V \\

. \\\

^\. \ ^\

N. \. \

0\\

\^^ \ >.

\ N. ^v

^^\^s

^\\

^\^^\

v^N^N

^^ ^^\ ^

^$^

^-^^

:£S^.

^-^^

^.^£S<

^^■^^---^.^"*

-^ ^^/^^^^-^

■^ ^ ' -^

■"^ ^*^

--^Zl^

^^^T:^

— — __

"""—*— ^^...^^^^ ■■

;

____

~~ — - _^ __

Fig. 217. Tangent scale for the determination of the inclination (according to Werenskjold).

482 Ocean Currents in a Non-homogeneous Ocean

scale 1 : 500,000, then the vertical exaggeration n is 200 and one obtains for
isopycnals

and for isosteres

1-885 ^ , , ,
v^-vi= ^^ 2:y (cm/sec).

4. Ekman's Theory of Density Currents Including Friction

Consideration of frictional effects in a stratified ocean is more difficult than in a
homogeneous sea for two reasons.

First, the mathematical difficulties increase considerably, and secondly, the depen-
dence of the frictional coefficients on the stratification is very incompletely known.
In a stratified ocean friction should be less than in a homogeneous sea and the intro-
duction of a constant frictional coefficient, which must be made, does not fit so well
under these conditions as in the case of homogeneous water.

Nevertheless, the results obtained on this basis afford some insight into the effect
of friction on the formation of density currents. Ekman (1905, 1906) has also dealt
with this in his theory of ocean currents and has made important contributions to
clarify this problem. A general solution, however, cannot be given. By means of some
typical cases only can conclusions be reached, from which the effects of friction can
be deduced by comparison with the frictionless cases.

A simple case is that where the specific volume decreases uniformly with depth and
the isobaric surfaces are thus inclined planes. If, as a consequence of this assumption,
there is no pressure gradient at a particular depth d (horizontal isobaric surface),
then taking

- ~ / = -fV and - - / = + fV
p dx ■' p dy ■'

(U, V are the components of the geostrophic current) the equations of motion
(XII 1.28) give

Z)2 d^u Z)2 8^v

o^ ^1 + ^ = ^ ^^^ o^ ITS

+ y=F and ^z tt^ - « = - ^. (XV.IO)

Therein D is the frictional depth (equation XIII.26). For a co-ordinate system with the
X-axis parallel to the isobaric surfaces (F = 0) and taking as before U = b (d — z) a.
solution can be given for (XV.IO). The velocity profile can be calculated for different
values oi djD (Fig. 218) from the very complicated equation obtained. The velocity is
given in the diagram in units of f//5; they can also be considered as given in cm/sec if
the total layer from the sea surface down to the layer of no motion d, of the dynamic
section oriented in the direction of the gradient, contains in each 1 km layer a total of
10^cusin(/> solenoids (for 45° there are 51-6 solenoids). The difference from the
velocity profiles presented in Figs. 173 and 174 for a homogeneous mass structure is
considerable. Wherever the depth of no motion d may be, the motion there occurs
nearly in a plane. The friction affects principally the direction of this plane. Table 1 36
gives the largest (amax) and the smallest (auxm) angle of deflection from the gradient

Ocean Currents in a Non-homogeneous Ocean

483

X, cm/sec

Fig. 218. Velocity profiles in density currents for shallow ocean depths (according to
Ekman). The unit of the velocity scale is U:5.

Table 134. Frictional influence on density currents in different depth of the ocean

0-25

0-50

1-25

2-50

"max

26"
26°

67"
62"

93"
82^

91"
86"

90"
90"

"surface ^^

100

direction and in addition the velocity of the surface current as a percentage of the
geostrophic current U. The vertical velocity decrease is at first very slow and then
becomes almost linear. By this it is shown that the law of parallel fields also applies
to a close approximation when frictional effects are present.

Simple mass distributions such as these rarely occur in nature. In addition Ekman has
also investigated cases in which the eflFect of a homogeneous solenoid field is superim-
posed on a gradient current. A lighter stratified top layer spreads out over a homogeneous
deep water. The lighter water body may, for instance, be coastal water lying in a wedge-
form off a long coast and can be regarded as a mixed layer of fresh water from the land
and of deep water. External forces are not taken into account ; at the boundary surface
between the top and the deep layer the water movement of the upper density current
exerts a shearing force on the deep water which gives rise to an "internal drift current".
A closer examination of the case of a boundary layer at a depth d, parallel to a straight
coast between a homogeneous upper and lower layer, gives the velocity profiles
for different values of dID presented in Fig. 219. The points on each curve refer again
to the depths 00, 0-1 D, 0-2 D . . . , below the sea surface. The part of the curve re-
ferring to the top layer is shown by a thick line ; the points on the thin part of the curve
(deep water) have been omitted for clarity. The unit of velocity is the same as in Fig. 218.
If the depth of the top layer is small as compared with D, there will be a strong deflec-
tion of the upper current away from the coast. The effect of the deep water lying just
underneath the top layer varies according to variations in the depth of the top layer.
U d < hD the deep water will in part be dragged out to sea by the water of the top
layer so that underneath this there will be a current directed away from the coast and

484

Ocean Currents in a Non-homogeneous Ocean

Fig. 219. Vertical structure in a convection current off a long straight shore (x-direction)

for a homogeneous top layer of the vertical extent d and homogeneous deep water (D,

frictional depth; unit of the velocity as in Fig. 218, according to Ekman).

only below this, the current is directed towards the coast. If, on the other hand,
d> D, then there will be a normal gradient spiral in the top layer and a corresponding
inverse one in the deep water. If the water of the top layer is stratified, the general
current structure will be significantly changed (Fig. 220). Now the deep water will be
carried along, only to a lesser extent. The deeper the surface layer, the closer will the
flow parallel the coast and the lesser will be the eff'ect on the layer beneath. As in the
case of Fig. 218 the current is limited to the stratified top layer and its intensity falls
near the boundary layer almost to zero.

^=0-25£?.

d--0-5L

d=0\D

/ y

-AZ'bD

Fig. 223. The same as in Fig. 219 for a stratified top layer (according to Ekman).

Ekman (1928 6) summarized these results and arranged them in a clear manner in
Fig. 221. Three alternative assumptions have been made on the thickness (in metres)
of the top layer d^\

(1) the top layer is divided into two homogeneous halves with a discontinuity
surface in the middle ( — x — x — ^);

(2) the top layer is stratified so that in it a density current is generated with a velocity
distribution following a cosine-function ( — • — • — ) ;

(3) in the top layer the velocity decreases linearly with depth and there is a dis-
continuity layer ( — o — ^o — ^).

Velocity profiles for the currents produced are shown on the right-hand side of
Fig. 221 ; in the upper picture for a top layer the thickness of which is assumed equal

Ocean Currents in a Non-homogeneous Ocean

485

to the frictional depth and in the lower layer is assumed as equal to double the fric-
tional depth. The thin hnes refer to the lower layer and the thick lines to the top layer.
The two arrow-heads at the right-hand edge connected with the + sign represent the
vector of the surface current in the case of frictionless motion. For sharper discon-
tinuity surfaces and a greater thickness of the top layer the velocity profile, as before, is
made up of two Ekman spirals. If the top layer is stratified there is in both cases a

Fig. 221. Density currents in a top layer considering friction and for motionless deep

water (according to Ekman).

current of almost uniform direction and the current velocity will decrease almost
linearly with depth. Due to the stratification of the current, intensity in the lower
layer (internal drift current) will be strongly reduced, and for a deeper top layer this
current will disappear almost entirely. The transport in the deep current will then be
insignificant. This leads to the important conclusion that: the sea surface under the
influence of external disturbances will adjust itself in such a way that the pressure gradient
arising from density dijferences in the top layer has a inaximum value at the sea surface,
decreases with depth and will largely or entirely vanish at the lower boundary of the
top layer; the deep water will remain practically motionless.

The "elementar" current in a vertically comphcated stratified ocean consisting of a
stratified top layer and an almost homogeneous deep water will thus, according to
Ekman, have the following three current constituents.

(1) The physical sea level and the isobaric surfaces of the top layer will be turned in
such a way that the pressure gradient has the same direction everywhere and will be
proportional at every level to the density; in the homogeneous deep water, however,
this pressure gradient will remain constant. The current produced by this mass structure
will be a simple gradient current.

(2) If the physical sea level and the isosteric surfaces are brought back to the initial
position, then an additional current resulting from this mass displacement adds to
the gradient current described above. This is called the density current.

(3) In addition, the effect of the wind on the sea surface generates a pure drift
current. This current will differ only slightly from that in a homogeneous sea if the top
layer is sufficiently thick. However, the density current will not be confined to the top
layer alone, but when this is reasonably thick, the influence on the homogeneous deep
water from above remains small.

486

Ocean Currents in a Non-homogeneous Ocean

Laboratory experiments with stratified water have been made by Sandstrom
(1908, 1918) in order to demonstrate experimentally the effect of stratification on
wind-generated currents. In the experiment, an air flow over the surface of a multiple-
stratified water mass in a narrow rectangular basin immediately produces a current in
the direction of the wind. The piling up of water at the windward end of the basin
gives rise to a counter current in the lower part of the uppermost layer ; there is a
closed circulation in this layer. Friction then produces a somewhat weaker circulation
with an opposite sense of rotation in the layer immediately beneath the uppermost one.
Further circulations are formed in successive layers beneath this, each with the
opposite (direct or indirect) rotational sense to that above it. Sandstrom's experimental
results for a narrow basin cannot be applied directly to actual conditions in the ocean.
In the laboratory experiment, in the first place, boundary conditions at the outer rim
of the narrow basin will play a decisive role, and secondly, the deflecting force of
earth rotation will have no effect and thus it is precisely that factor which most
decisively influences ocean currents in nature that is left out of consideration. The
laboratory experiment is thus apphcable in nature only to narrow confined sea basins
and to lakes.

5. Oceanographic Applications of Bjerknes's Circulation Theorem

The theory of ocean currents in a non-homogeneous sea received a very strong
stimulus from the circulation theorem of Bjerknes, since it opened the road for studying
in a quantitative way and for the first time the effects of baroclinic mass fields. There
are manifold possibilities to apply this theorem in oceanography some of which will
be discussed here in more detail.

{a) The Steady State of Motion

The most important use of the equation (X.54) is for the steady state in which the
circulation accelerations vanish. In this case

N=f

dfn

(XV. 11)

(here again A'^ = number of solenoids, / = Coriolis Parameter, F^ = area of the
projection of curves on the sea surface). The curve s is now made up of the two
station verticals AC and BD and of two isobars AB and CD (Fig. 222). The water

Ocean Currents in a Non-homogeneous Ocean

487

masses at the upper level move with an average velocity ^o and those at the lower level
with an average velocity Dj at right angles to the section. After unit time the water
elements, initially at AB, will lie at the line A'B' and those from the isobaric interval
CD at CD'. The total surface ABCD transforms into A'B' CD'. The change of the
projection of the surface ABCD on the sea surface thus becomes A'B'C'D", so that
ciF^ldt = {vq — v^L, where L is the distance between the two stations A and B.
Equation (XV. 11) combined with (X.45) gives

{Vo = t'l) =

Da- Di

(XV. 12)

This equation, which was first derived by Helland-Hansen (1905), forms the
fundamental equation of dynamic oceanography. From the difference in dynamic
depth of the isobaric surfaces Da — DbS. simple calculation gives the increase in velocity
from one surface to the next. Analogous treatment to that on p. 466, however, affords
only velocity differences and only the component at right angles to the selected section
is obtained. Equation (XV. 12) contains fundamentally the same as equation (XV.7)
derived directly from the equations of motion. In the practical appUcation of (XV. 12)
it should be noted that /)„ — Di, has to be expressed in units of the potential, that is,
in dynamic decimetres when the metre is taken as the length unit. The difference in
dynamic depth anomaly, e^ — €{,, can, of course, be used instead of the difference
Da - D,.

The section to the south of the Newfoundland Banks between stations 205 and 206
can be used again as an example (see Fig. 202). Table 135 contains the dynamic depths,
their anomalies and values of €„ — ^6 for selected pressure surfaces down to 750
decibars. In equation (XV.12) <^ = 41° 10' N.;/= 9-60 x 10-^; L = 59 km and the
denominator is 5-664. The anomaly differences are multiplied by 10 in order to obtain
dynamic dm ; this gives then v in m/sec. The last column gives velocities on the assump-
tion that there is no motion at 750 m (see Table 133). If calculations of this type are
available for a sufficient number of station pairs it is possible to obtain a complete
velocity field at right angles to the cross-section. A comparison of the velocities cal-
culated in this way from the mass field with the observed velocities was first given by
WiJST (1924) for a cross-section through the Gulf Stream in the Florida Strait. The

Table 135. Computation of the velocity profile south of the Great Banks of

Newfoundland.

St.

205

i St.

206

Pressure
(dbar)

D Du

Depth

Anomaly

1 Depth

Anomaly

t^a ^b

(cm/sec^)

(cm'sec)

(dyn. m)

0 —

0—

48-68875

006225

48-7175o

009 lOo

002875

5-1

125

121-69188

0-14676

121-75713

0-2120i

0-06525

11-5

250

243-2725o

0-2525i

243-40776

0-3877^

0-13526

23-9

450

437-5985o

0-3650i

437-84276

0-6092,

0-24426

431

750

728-7215o

0-5020i

72908476

0-86527

0-36324

64-1

488

Ocean Currents in a Non-homogeneous Ocean

agreement was very satisfactory; later this kind of comparison has often been repeated
confirming the results.

If, instead of as in Fig. 222, the vertical section is placed in the direction of the
relative velocity Vq — V^, then there will be no component at right angles to the surface,
that is, in (XV. 12) ^o — i^i = 0 as well as £)« — D^ = 0 and the dyn. depths in the cross-
section must be the same at C and D. If one of these verticals is kept fixed, then the
other will move away at the relative velocity Fq — V^ and for every point along its
track always applies Da — Dt, = 0. This implies that: curves of equal dyn. depth,
which then give the dyn. topography of an isobaric surface relative to another, represent
at the same time stream lines of the relative velocity {velocity of one surface relative to
that of the other).

This theorem is of great importance in the discussion and interpretation of the
relative topographies of individual pressure surfaces in the ocean. An example is
presented in Fig. 223 which shows the relative topography of the isobaric surface at
750 decibars for the same area containing the section shown in Fig. 202. The indication
arrows show the direction and the intensity (nautical miles per hour) of the (relative)
velocity of the layer at 750 m depth relative to that of the surface. If the water in this
depth is motionless, then they represent the sea surface current. The dyn. isobaths
are stream lines for the whole system.

.57°W 56

'W 56"

Fig. 223. Dynamic topography of the 750-decibar surface south of the Great Banks of

Newfoundland according to the observations from 5 to 7 May 1922 (according to Smith).

The arrows indicate the computed relative current in nautical miles per hour.

Both applications of the circulation theorem have made use of curves in vertical
planes, which contain a large number of solenoids. The theorem may also be applied
to horizontal curves, which include little or no solenoids. For curves of this type the
first term on the right-hand side of equation (X.54) vanishes and there remains only
the term expressing the effect of the Coriolis force. On integration it gives

-Co=-/(F,

Fm 2).

(XV. 13)

Ocean Currents in a Non-homogeneous Ocean 489

A horizontal circulation free-curve (Cq = 0) will acquire by contraction a cyclonic
circulation and by expansion an anticyclonic circulation.*

If curves extending as parallel circles all around the Earth and containing an ocean
covering the entire Earth are carried towards the equator by the general oceanic
circulation, then their projection on the equatorial plane will expand and they will
thus acquire a zonal anticyclonic circulation, that is, from east to west. On the other
hand, if they are displaced towards the poles there will be a shrinking of the areas
enclosed within the parallels and thus there will be a zonal cyclonic movement from
west to east. Considerable changes can also occur in the area enclosed by horizontal
curves flowing over a submarine ridge thereby causing the formation of cyclonic or
anticyclonic circulations. These will be superimposed on the basic current and will
give rise to a wave-form character of the current structure (see p. 431).

(b) The Sandstrom Theorem

In the ocean there exist closed circulations of greater or smaller extent, which are
maintained by the continuous supply of heat at certain fixed places and the continuous
withdrawal of heat at others. These sources of heat and cold maintain the differences
in specific volume. Thus circulation velocity in a frictionless medium will continuously
increase, since the circulation acceleration in equation (X.44) has a positive value. In
reahty, however, all circulations are affected by frictional forces. Another term R
must therefore be added to equation (X.44) containing all the frictional effects. There
will be a steady state only when

- I adp-\- R^Q (XV.14)

that is, in a steady state (disregarding the rotation of the Earth) the work done by the
pressure forces (i.e. — /« « dp) is used exclusively in overcoming the frictional forces.
This can only be the case when

adp<0. (XV. 15)

From this controlling equation it is easy to draw conclusions as to how the sources of
heat and cold should be located in space inside the circulation in order to allow for a
stationary state. The concept of sources of heat and cold must be given in the ocean a

* It follows from equation (X.54) dCjdt = fcIF^dt that for an increase in the area dFJdt > 0
there will be an anticyclonic deflection and correspondingly for a decrease a cyclonic deflection.
This can, of course, also be derived directly from the equations of motion. For a geostrophic friction-
less current these are

\ cp \ cp

— /f = ^— and + /m = 7— .

p (}x p cy

By cross-wise differentiation and rearrangement

/div vh + Pv = 0, whereby j3 = -r .

If the current is divergent (div r^ < 0) y must be positive; this indicates that the deflection will be
anticyclonic or to the right in the Northern Hemisphere ; for a convergence (div vh > 0) u is negative
with a corresponding cyclonic turn.

490

Ocean Currents in a Non-homogeneous Ocean

somewhat wider sense. In the real ocean differences in specific volume are produced
not only by h,eat gain or heat loss, that means thermally, but also by changes in salinity.
Evaporation will increase salinity and precipitation, ice melting and the inflow of
fresh water (run-off) will reduce it. An increase in salinity has the same effect as a
cold source and a decrease in salinity will be equivalent to a heat source. In the
following the sources of heat and cold will be taken as including always the combined
effects of both factors.

In a Camot cycle one single and complete revolution shall now be considered on an
[a,/j] -diagram (Fig. 224) consisting of two isobars {dp = 0) and of two adiabatic
curves along which there is no addition or removal of heat and changes will occur only
due to expansion or contraction. There are two possible cases:

Fig. 224. Camot's cycle. Case o: heat source at lower pressure (small ocean depth) than cold

source. Case b: heat source at higher pressure (great ocean depth) than cold source. A

stationary circulation is only possible in case b, not in case a.

(a) Clockwise cyclic process. From 1 to 2 at a constant, but lower pressure {p^ < p^,
in the upper part of the sea) there will be a heat input (heat source), from 2 to 3 there
will be an adiabatic compression followed from 3 to 4 by a heat removal (cold source)
at higher pressure (in the lower part of the sea). Finally, an adiabatic expansion occurs
from 4 to 1. Evaluation of the integral (XV. 15) gives, since the isobaric sections of the
cycle make no contribution

a dp = \ (a

a4.i) (ip > 0»

since both (02,3 — 04,^) as well as dp are greater than zero. The pressure forces are
incapable to do work. Any existing circulation will in time be destroyed by frictional
effects.

(b) Counter-clockwise cyclic process. The heat source works at high pressure

I < P2, in the lower part of the sea).

In this case

a dp

(04,1 — aa.a) dp < 0.

The pressure forces are capable to do work. If this is so large as to overcome all the
frictional forces there will be a steady circulation.

If there were no friction, this would be a reversible process and the degree of
efficiency of this thermodynamic machine would be given by W = {Q^ — Q-^iQi,
where Q^ is the amount of heat absorbed by the medium from its surroundings at the

Ocean Currents in a Non-homogeneous Ocean

491

heat source, and on the other hand Q^ is that lost to the surroundings at the cold
source. If frictional effects are present, then the process will be irreversible. The
machine will give off a quantity of heat Qo, during the course of this process which is
greater than in the reversible case {Q'c, > Q2). The degree of efficiency of such a
circulation is less than Wand is given by (Qi — Q'^IQ^. In a circulation for which the
work done by the pressure forces is exactly sufficient to balance the loss of energy by
friction the degree of thermodynamic efficiency will be exactly zero. The Sandstrom
theorem thus states: a closed steady circulation can only be maintained in the ocean if
the heat source is situated at a lower level than the cold source. Sandstrom (1908) in
order to elucidate the content of his theorem has performed a number of very instruc-
tive laboratory experiments. Later on Bjerknes (1936) has presented a detailed analysis
of all the questions raised when dealing with thermodynamic machines of this type.
The two most important of the Sandstrom experiments are:

{a) Heat source at a higher level than cold source. Here a single water type is con-
tained in a narrow basin but there are two sources (Fig. 225, upper picture). The heat
source ("warm") lies at a higher level than the cold source ("cold"). At the beginning

Worm

Cold

^ . ► ^ - — >- — ^

Cold

^ '^ ^^^^z^

Fig. 225. Upper picture: heat source situated above cold source : no circulation and vertically
stable stratified water layers. Lower picture: heat source situated below cold source; genera-
tion of a stationary circulation in the layer between the levels of the heat and cold source.

of the experiment motions will be set up because the heated water will rise in the layers
above the level of the heat source and cooled water will sink in the water layers
below the cold source. However, when the upper water reaches the temperature of the
heat source and the lower water that of the cold source, these water movements will

492 Ocean Currents in a Non-homogeneous Ocean

cease and there will be a stable stratification with the temperature decreasing with
depth. A state of no motion is created since the circulations previously present will be
halted rapidly by friction.

(b) Heat source at a lower level than cold source. This is the same experiment as in
(a) except that the position of the two sources is inverted. Convectional currents will
be set up in this case also, but soon there will form a steady circulation confined to the
layers between the levels of the two sources (Fig. 225, lower picture). Above there will
be a water movement from warm to cold and below from cold to warm; the most
heated water will be above the level of the heat source and the coldest below the cold
source. But these layers will not take part in the circulation which is solely confined to
the intermediate layers.

Later on, Sandstrom modified the experiment in several ways, especially to show
more clearly its application to oceanographic conditions; basically, however, these
do not give any new results. Jeffreys (1925) has questioned the general validity of
Sandstrom's conclusions but Sandstrom's deductions from the circulation principle
are undoubtedly correct. The circulations produced by thermo-haline differences are
the more intense the greater the vertical distance between the level of the warm and that
of the cold source. However, conditions existing in nature in the ocean are not parti-
cularly favourable to the formation of any more intense circulations of this type,
since the principal heat supply in the ocean is primarily due to the combination of
solar radiation and back-radiation from the atmosphere and the loss of heat primarily
due to outgoing radiation. These processes operate to a very large extent at the
boundary between the ocean and the atmosphere (almost horizontal sea level and
evaporation and precipitation also act here. The vertical distance between the location
of the heat and cold sources is thus very small. Probably the heat source in equatorial
areas lies somewhat deeper than in higher latitudes, but nevertheless the thermo-
haline circulation must be limited to a very shallow top layer. Observations provide
complete confirmation of the consequences deduced from the circulation principle
(see p. 576).

6. The "Reference -level" for the Conversion of the Relative Topography of the Press-
ure Surfaces into the Absolute One

The relative topography of the isobaric surfaces (relative to the sea level) assumed
as plane) can be determined by the methods described on p. 309 and the following
pages. Using equation (XV. 12) this also gives the relative velocity differences from layer
to layer. In order to obtain a complete quantitative knowledge of the water move-
ments it is necessary to convert these relative topographies into absolute topographies.
This can be done if the relative topography can be referred to a known topography
of any isobaric surface. This determination of the absolute topography would be
easy if it were possible to determine from current measurements such a depth level at
which the velocity of the current is zero, since at this "depth of no motion" the isobaric
surface must coincide with a level surface ("Niveauflache").

In this way, for example, Wiist used the current measurements made by Pillsbury
in the Floriaa Strait in oraer to determine the current profile of the Gulf Stream from
the mass field. The number of current measurements available for the open ocean is,
however, insufficient to fix with some accuracy the position of such a "zero level"

Ocean Currents in a Non-homogeneous Ocean

493

("Nullflache"), quite apart from the fact that short series of current measurements are
almost always strongly disturbed by the tides. Thus the essential data needed to
decide about the position of the "zero level" is largely lacking. The effort to utiUze
the observations as fully as possible and to determine the pressure differences as good
as possible, at least in the upper layers, has led to place the zero surface as deep as
possible. This choice was also suggested by the generally rapid decrease in the velocity
of the currents with depth.

Over the entire area under consideration most investigators have thus usually placed
the zero level at a constant dynamic depth and as deep as possible (as far as the water
depth and the observations available allowed), and from this have derived the absolute
topography of the pressure surfaces and that of the physical sea level from the relative
topographies. Table 136 presents a summary of all the depths selected for the zero
level by different investigators. The differences of more than 1000 m indicate that these
are pure assumptions for which there is no firm basis. However, all investigators have
been aware of the inadequacy of this procedure and have regarded the selection made
purely as a make-shift. The assumption of a zero level at a constant large depth will,
of course, conceal all currents in the layers just above and below this depth, and these

Table 136. Depth of the ''zero level" {Nullflache'') in the Atlantic Ocean according to the
assumption of difl'erent investigators

depth

Investigator

Year

(m)

Investigator

Year

(m)

Bouquet de la Grye .

1882

4000

Helland-Hansen and Nansen

1926

2000

Mohn

1885

550

Jacobsen ....

1929

1000

Zoppritz

1887

2000

Iselin ....

1930

1200

Wegemann

1899

1000

Helland-Hansen

1930

1000

Schott ....

1903

500

Iselin ....

1936

1800

Castens ....

1905

650

are thus falsified if by chance the zero level selected does not correspond with the
actual position of such a level. On the other hand, the deeper the zero level is placed,
the less will it disturb the pressure conditions at the sea surface.

To obtain a correct idea of the deep current, it is not sufficient to assume a constant
depth for the zero level. Such an assumption, moreover, does not correspond to the
dynamics of the ocean currents in nature and, as has been stressed by Ekman (1939)
takes no account of the topography of the sea bottom. These problems of dynamic
oceanography have been dealt with by Dietrich (1937 a, c), who has thrown light on a
number of aspects of them. The zero level, more suitably could be called "reference-
surface" and has to be placed at such a depth where the velocity component at right
angles to the dynamic section under consideration is zero. It must, of course, closely
adapt to the mass structure of the entire oceanic area, since this is in fact a conse-
quence of the currents and is closely connected with them. In these circumstances it is
to be expected, especially when larger areas of the sea are taken into consideration,
that the reference-level for the reduction of relative into absolute topography must be

494 Ocean Currents in a Non-homogeneous Ocean

a surface of locally varying depth. The determination of its form and the different
factors that must be considered for fixing its position in oceanic space is not an easy
task. It should be stressed that the choice of such a surface is always more or less
subjective, and such an assumption can only be made plausible by giving proper
weight to all the different view points which are in question.

{a) Determination of the Topography of the Reference-Level

A first attempt was made by Dietrich in an investigation of the dynamics of the Gulf
Stream to introduce a reference-level of variable depth by investigating characteristic
features in the distribution of oxygen in order to fix the reference-level. He thus
accepted the widely held view that the layers showing the intermediate oxygen minima
(see Pt. I, p. 66 and following pages) are at the same time also layers of very weak
motion or of no motion at all, and could thus be regarded as motionless boundary
layers between individual components of the deep-sea circulation. However, Rossby
(1936 a), ISELiN (1936) and especially Wattenberg (1938) and Sverdrup (1938 M
have questioned this assumption and have expressed doubts about the suitability of
these oxygen minima as reference-levels. In the upper layers of the ocean the oxygen
distribution can be regarded, on the one hand, as a consequence of thermal and bio-
chemical oxygen consumption, and on the other hand, of the renewal of the water
masses by horizontal advection. The intermediate minima are thus regions of parti-
cularly strong oxygen consumption and can hardly be regarded as completely motion-
less layers. The results obtained by Dietrich for the currents in the Gulf Stream on the
basis of this assumption are not such as to give confidence in reference-levels derived
from the oxygen minimum. Even the customary division of the water masses of an
ocean, pictured by major longitudinal and transverse section> and allowing for the
characterization of the different water bodies, is scarcely suitable for the determination
of the topography of the reference-level. Even though they may be practical and useful
in giving a general qualitative picture of the meridional and zonal velocity components
of the ocean currents.

Defant (1941 b) has gone a quite different way in order to determine the dynamic
reference-level in the Atlantic, which avoids the use of any particular boundary layer
between the individual water types and makes use only of dynamic evaluations of
observational data, which must be closely connected with the structure of the water
masses of the particular area. The differences in dynamic depth of the pressure values
between two neighbouring stations give, by means of equation (XV. 12), a relative
measure of the velocity difference perpendicular to the cross-section between the sea
surface and the corresponding depth. When these differences are plotted in an appro-
priate co-ordinate system (ordinate :pressures; abscissa :difference in dynamic depth)
they give a relative vertical velocity profile at right angles to the section between the
two stations (Fig. 226). This profile cannot be converted to an absolute velocity profile
without knowing the zero point on the abscissa. By comparison of a large number of
difference-curves for neighbouring pairs of stations it shows in most cases that in
each profile there is a layer of considerable vertical thickness in which the differences
in dynamic depth are constant or almost constant. If the zero point of the abscissa
scale is placed outside of this layer then the entire layer must have a constant velocity.

Ocean Currents in a Non-homogeneous Ocean

495

0
400
800
1200
1600
2000

dyn.cm
-12 -8 -4 0 +4 +8

-4 0 +4 +8

dyn.cm

+ 12

Fig. 226. Schematic example for fixing the reference-level by means of the vertical distribu-
tion of the dynamic depth of the standard pressures of two neighbouring stations. (The
lower "displaced" scale of the abscissa only has its correct position, if the reference-level is
assumed in the layer denoted by the vertical arrow; a position of the reference-level at the
dashed arrow, for example, would be quite improbable.)

while the dynamic structure of the other layers will be divided up in a rather unintelli-
gible way. It is more plausible to suppose that this more prominent layer should be
motionless, or almost motionless, so that the reference-level should lie within it.
Such a layer with obviously low velocities is apparently characteristic not only for the
pair of stations under consideration, but is to some extent depending on the pressure
field of the entire oceanic region under consideration. The reliability of this method is
increased if the individual reference depths, determined from a large number of station
pairs, can be combined to give a closed system representing a definite topography of
the reference-level.

To illustrate the method the difference-curves for the dynamic depths are shown in
Fig. 227 for a meridionally distributed set of stations in the Atlantic ; for each curve the
vertical extent for which a layer of no motion or only weak motion is most probable,

dyn cm

0 1 B 12 16 20

dyn cm

Fig. 227. Fixing of the dynamic reference-level for a series of meridionally distributed
stations in the Atlantic Ocean.

496

Ocean Currents in a Non-homogeneous Ocean

110° W 100° 90'

100° 90° 80° 70° 60° 50°40°30°20° 10° 0° 10° 20° 30° 40° 50° 60° E

Fig. 228. Position of the reference-level for transforming relative topographies into
absolute depth (numbers in 100 m units).

Ocean Currents in a Non-homogeneous Ocean 497

is marked with a vertical double arrow. The reference-level for conversion of relative
into absolute topography must lie within this layer. Already these station pairs show
roughly the meridional distribution of the depth of the reference-level in the Atlantic :
lower depth in high latitudes (approx. 1500 m or deeper), rising up to 500 m at the
equator. The topography of the reference-level can thus be derived for the whole of
the Atlantic from a large number of such diagrams. Figure 228 presents the topography
determined by this method. The lines are drawn at 100 m intervals (or decibars);
for a reduction of the relative into absolute pressure values it is sufficient to know the
position of the reference-level to the nearest 50 decibars. It is clearly shown that the
assumption of a reference-level of constant depth can never do justice to the dynamic
structure of the water masses of the Atlantic Ocean; even over smaller oceanic areas
there are appreciable variations in its position. Along each meridian the depth of the
reference-level is least near the equator (up to 400 m), in the Southern Hemisphere it
sinks uniformly to great depths in high latitudes. But in the Northern Hemisphere
conditions are more complex. From the equator it sinks at first to a secondary mini-
mum between 5° and 10° N. (about 900 m), then rises again to another maximum
between 10-20° N. and from there begins the lowering towards the north-west to
greater depths. The irregularity in the northern subtropics has the same form as the
asymmetry in the position of the subtropical and tropical thermocline (see Pt. I,
p. 120). There is undoubtedly a causal coimection between the two phenomena. In the
Gulf Stream region there are considerable deviations from normal. Near to the
current core (intense flow) the reference level rises steeply upwards to a depth of
1000 m or less. This phenomenon, which belongs to the characteristic features of this
area, must be connected causatively with the inclination of the isosteres in a stratified
ocean with intense motion (see p. 331).

From the chart shown in Fig. 228, Neumann (1954, 1955) has computed zonal
averages of the depth of no meridional motion D (zero level) for the North Atlantic
and has plotted them against the latitude (Fig. 228 a). Individual values along the
20°W-meridian were used for the South Atlantic, since the variation in D in the east-
west direction is small as compared with the variation of D in a meridional direction.
In Fig. 228 a the values of D are marked by circles and the full drawn curves represent
the function

D= - K?,mcl>^- Kcos &. (XV.16)

The constant ^ is different in the Northern and Southern Hemisphere but the increase
of D with latitude follows this function closely except in the equatorial regions, where
apparently another physical law applies (see Pt. I, p. 120).

Excluding the equatorial regions, the relative variation of D with latitude is given by

15 'I = - '^"*- "^^•'"

Then, it follows from the Coriolis parameter, f=2w cos d that

1 df

-f-^^-- tan^. (XV. 18)

498

N 60°
0
D
(mJ!

K>00

2000

3000t:

Ocean Currents in a Non-homogeneous Ocean

50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50°

1000

2000

3000

60° S
0
D
(m)

1000

-2000

-3000

Fig. 228a. Average depth of the reference-level (layer of no motion) in the Atlantic Ocean

(according to Neumann)).

Thus for the large scale major oceanic circulations the fundamental relation

8^ D 8&

(XV. 19)

is obtained. An investigation of the reference level (layer of no motion) similar to that
made by Defant was also carried out by Neumann (1942, 1943) in an evaluation of
observational data for the Black Sea. For the strong vertical stratification of this
adjacent sea the topography of the reference level is more closely connected with the
position of the boundary layers characterizing this vertical structure. Figure 229 shows
the position of the different boundary surfaces in a longitudinal section near 43° N,
It is almost the same everywhere: the lower plankton limit, the maximum density

40° E 41=

Fig. 229. Depth of different characteristic boundary layers in a longitudinal cross-section
through the Black Sea in 43° N. (according to Neumann).

Ocean Currents in a Non-homogeneous Ocean

499

gradient, the upper limit of the H^S-\a.yQV and the reference level are all more or less
coincident (except near the coastal areas in the eastern part). All these surfaces join
here, forming a single closed system, an almost motionless boundary layer.

If in an adjacent sea a density discontinuity layer is found everywhere, the deter-
mination of the position of a dynamic reference level is considerably simplified, since
the lower limit of the top layer is then usually also the lower limit of the upper flow
and the discontinuity layer coincides with a layer of no motion. These methods have
already been used by Witting (1918) in his investigations on the continental rise
around the Baltic. This simple method can, of course, only be used when the thickness
of the top layer is not too great ; it is also possible to apply this method with success to
shelf areas, having a sharp subdivision in the vertical into two layers.

A new method for the determination of the depth of no meridional motion has been
presented by Stommel (1956). It is of interest in so far as it permits a determination of
this depth directly from the observed vertical distribution of the oceanographic
factors, and because it also shows that there is in actual fact no depth of no motion in
the ocean but rather the depth of no meridional motion always coincides with the
layer of maximum vertical velocity. From the general equations for a wind driven
motion and the continuity equation cross differentiation leads to the following three
relations :

(XV.20)

The quantity pv in the third equation can be ehminated by means of the first equation,
giving

8\p^^^ ^g 8p 8^

g dp 1

fcx f

^ 1 X

8z^ '

1 (^") =

g op 1 8^T^
f cy f 8z^ '

- (ph) =

0'^

^ 1 8

-r PV — -> ^^
/ f 0^

where

8z^

f(=)

/2 8x 8z^

PCV.21)

8
8x

(7-) - If)

This function F(z) is more or less indeterminate, but accord'ng to Ekman differs from
zero only in a thin upper layer extending from r = 0 to the depth of frictional influence.
F(0) is known in terms of the distribution of the wind stress on the sea surface. If
the sea bottom is at —d, then the first integral of equation (XV.2] ) can now be obtained :

Sz^P^'^^-f

^(-) + C

(XV.22)

whereby (f'(r) is defined

500 Ocean Currents in a Non-homogeneous Ocean

and where C is an integration constant. The meaning of the function 0(z) is easily
understood, since for a purely geostrophic flow (from the first equation in XV.20 it)
follows

pv = 0(z) + C.

The constant C is the indeterminate reference velocity and the determination of C
can be readily seen to be equivalent to the determination of the depth of no meridional
motion, that is, the depth at which pv vanishes. Since for deeper layers F = 0, it
follows from (XV.22)

^ (PH') = 0.

By this it is shown that the level of no meridional motion coincides with the level of
maximum vertical motion. Since the bottom currents are rather weak, the hypothesis

IFz

allows the integration of equation (XV.22) between z and —d. Taking F{ — d) = 0
and p\v{ — d) ^ 0, the following expression for pw is obtained

pw = J

0(z) dz + C. (- + d)

F(z). (XV.23)

At the surface, r = 0, pw vanishes; the quantity F(0), according to (XIII. 27) is the net
convergence of the wind-driven layer and (XV.23) gives

■^ F(0) - [" jHz) dz

The depth at which <P(z) + C vanishes, is the depth of no meridional motion.

In physical terms the method, given in formal terms above, can be loosely described
in the following way. At any geographical position in the ocean the distribution of
the winds produces a net convergence (or divergence) of the surface waters. In the
steady state the only outlet (or inlet) for this water is downwards (upwards) through
the bottom of the frictional layer. In the deep frictionless (by hypothesis) geostrophic
flow, water elements will stretch (or shrink) vertically as they move towards the poles
(equator). The cumulative effect of this expansion or contraction added up over the
entire vertical column from the ocean bottom to the bottom of the frictional layer
leads to a vertical component of velocity which, by the conservation of mass, must be
equal to that induced at the bottom of the frictional layer by the winds. This balance
will only hold for a specific choice of the reference-level which thereby fixes this level
(Stommel).

Stommel has given a numerical example for two "Atlantis" stations situated at
about 32° N., 50^ and 63'' W., respectively. Here the depth of no meridional motion is
found to be at about 1500 m; the maximum vertical velocity 24 x 10"^cmsec-\
also occurs at this depth. This depth agrees well with that inferred by Defant from his
method.

Ocean Currents in a Non-homogeneous Ocean

501

{b) Conversion of Relative to Absolute Topographies Using a Reference-level of
Varying Depth
For a reference-level of constant depth there is little difficulty in the conversion from
relative to absolute topography (see p. 21 1). The differences in dynamic depth represent
at the same time also differences in the physical sea level and in the level of individual
isobaric surfaces, respectively. If the depth of the dynamic reference level varies from
place to place this simple procedure can no longer be used ; the individual differences
of dynamic depths above or below the reference level must be coupled or inter-
connected one to the next in a suitable way in order to construct step wise the surfaces
of equal pressure (Dietrich, 1937 a). To determine the absolute topography of a
pressure surface Pq above the reference level, three oceanographic stations A, B, C
were chosen Sind Pa,Pb,Pc are the pressures at the points on the reference level at which
by necessity the pressure surfaces parallel the level surfaces (Fig. 230).

Fig. 230. To the method of transforming relative into absolute dynamic topographies.

If the stations are sufficiently close to each other, the dynamic reference level can
as an approximation by broken up into a step wise course. Along the section from
A to B the mean pressure will be given by

i (Pa + Pb)

and that between C and D by

Pa, 6

Pb, c = HPb+Pc)'

The dynamic height differences 8^ and S^ of the isobaric surface p^ over the mean
reference level can be determined in the usual way for stations A and B, as well as the
differences in dynamic height b\ and Sc of the isobaric surface Pq above the reference-
level between B and C. Then S„ + Sj, and S^ + S'b + Sc are the vertical deviations
of the isobar /7o from the level surface, running through the point A. The conversion
can thus be made quite simply for dynamic sections. If the stations are distributed over
a larger oceanic region, then the condition has to be satisfied that the absolute values
calculated along different paths (sections) must lead to the same value. Defant
(1941 b) has developed a triangle method which has been found very useful in the
determination of the absolute topography of the Atlantic Ocean from a large network

502

Ocean Currents in a Non-homogeneous Ocean

of stations. However, it requires laborious calculations since the errors occurring with
each triangle computation, although not large, must be eliminated by a smoothing
technique from triangle to triangle. Neumann has in some way modified this method
for practical use by taking all stations with the same reference level depth together,
thus obtaining a series of pairs of stations with constant reference level depth. For
these, however, the previous simple procedure is applicable. To connect one series of
station pairs to the next requires only one station triangle, and this can be selected in
the most favourable position where the triangle errors are small. In this way each
station series can rapidly be connected to the next with minimum error thus over
coming the difficulties otherwise occurring for a varying depth of the reference level.

In shallow parts of the sea the reference-level is usually found below the sea bottom
and the method described above can no more be used. It is, however, desirable to know
the absolute topography of the pressure surfaces in these shallow waters also, especially
as the most intense currents are often found here. Jacobsen and Jensen (1926), as
well as Helland Hansen (1934), have devised methods for calculation in this case.
It is necessary to preassume for these that the internal friction can be left out of con-
sideration, and the velocities as well as the horizontal pressure gradients at the sea
bottom should be zero. The method proposed by Helland- Hansen is based on the
following reasoning :

Figure 231 shows a dynamic section starting at a coastal point E across a shelf con-
taining station D, C, B and ending at station A out in deep water. The thin lines are

Fig. 231. To the method of fitting shelf stations together with deep-sea stations.

isosteres. In the sea between A and B the dynamic reference-level runs along the thick
dashed line. In the shelf area BCD the depth of the sea is less than the depth of the
reference-level in the deep water. In order to obtain the deviations of the dynamic
topographies of the isobaric surfaces we can imagine for the shallow part a fictive
vertical section from B to D below the level of the sea bottom, assuming the velocities
to be zero at the sea bottom. The isosteres in this imaginary section are horizontal and
therefore there is no motion in this part. The actual movements in the real section
and that in its imaginary extension will thus be the same. The latter can, however, be
used to extend the topography of the pressure surfaces above the shelf as far as the
coast.

Ocean Currents in a Non-homogeneous Ocean

503

In the method presented by Jacobsen and Jensen, further assumptions are
added to those used before which simpHfy matters even more. A and B (Fig. 232)
are two stations at which observations are available down to the bottom. The depth at
B is greater than at A and the difference in the physical sea level between A and B has
to be found. Aq and Bq are the points on the sea bottom at the stations A and B and

Fig. 232.

A 0^1 shall be a level line ("Niveaulinie")- The dynamic height difference BqB^ is denoted
by h and the specific volumes at .4o and B^ by aA,o and aB,i. Provided that:

(1) the sea bottom AqBq is linear in the vertical section and

(2) within the triangular section A o^o^i the mass field is linear and the isosteres are
therefore straight, equidistant and parallel lines and

(3) the pressure gradient vanishes at the bottom.

Then a simple integration method enables the required level difference to be cal-
culated by first calculating the difference in sea level between A and B, on the assump-
tion that the pressures at ^q and B^ are the same, and then adding the correction term
hK^Ba — oiA,o)- Ekman (1939) has shown that the method of Helland-Hansen leads
to exactly the same correction term. Both methods require that not only the current
velocity but also the horizontal pressure gradient should vanish at the sea bottom.
The first condition is satisfied because of the bottom friction, but the second is in
many cases a rather doubtful assumption, since considerable density differences
sometimes appear in both vertical and horizontal directions, at the shelf bottom.
Before applying the method it is thus first necessary to ascertain whether the pre-
sumptions are approximately satisfied or not. The method of Jacobsen and Jensen is
simpler for use and less time consuming than that of Helland-Hansen and requires also
less complete data.

A third method has been suggested by Sverdrup and co-workers (1942, p. 451).
They postulate below the sea bottom an imaginary water body in which the specific
volume a (or its anomaly S) and the slope of the isosteres is given at each depth by the
corresponding value on the continental slope. It is easily shown that the slope of an
isobaric surface pi relative to that of pa can be computed approximately from the
simple equation ip = — is(8i — So), where is is the mean slope of the S-lines between
Pi and p.2: Sj and S., are the specific volume anomalies at points 1 and 2. The mass
distribution in the imaginary water body then gives the pressure distribution, and the

504

Ocean Currents in a Non-homogeneous Ocean

method thus avoids the difficulty that the horizontal pressure gradient should vanish
at the bottom. Groen (1948) has somewhat modified this method by assuming that
only the slope of the isosteres in the imaginary water body is identical at each depth
with that at the bottom slope. The values of the anomaly at the bottom slope and the
distribution of the slopes over the entire space completely determine, however, the
entire distribution of the specific volume. No special assumption about the distribution
of 8 is required. The difference from the previous assumption cannot be very large but
the method is more correct. Therefore, all the methods described here give results which
essentially do not differ from each other.

7. Remarks About the Observational Material Necessary for a Dynamic Computa-
tion and Critical Discussion of the Procedure

In order to understand the importance of the absolute topographies of the isobaric
surfaces it must be realized that a knowledge of these allows a complete evaluation of
the field of motion at individual depths. The current vectors parallel in this field the
isohypses of the pressure surfaces, and the velocities are inversely proportional to the
distances between them as well as to the sine of the latitude. In the Northern Hemi-
sphere for an observer looking downwards along the slope of the pressure surfaces
(Fig. 233) the current will flow to the right. A practical formula for the numerical
evaluation of topographies is

„ = Jt_ f_ (XV.26)

2(x) sm cf) An

from which v is obtained in m/sec if AD is entered in dynamic metres and An in ordinary
metres.

Fig. 233. Schematic representation of an absolute dynamic topography with the corre-
sponding velocities and diagrams of forces (G, gradient force; C, Coriolis force).

The following prerequisites for the use of this formula should be borne in mind:

(1) the topographical charts must approximate the actual state within the particular
oceanic area at a definite time as accurately as possible,

(2) the currents must be steady,

(3) it must be possible to disregard the effect of friction.

Ocean Currents in a Non-homogeneous Ocean 505

At the present time this dynamic method of computation is used extensively every-
where. It is used particularly for the dynamic evaluation of widely varying vertical
profiles and gives information on the water displacements at right angles to the profile,
The scientific treatment of observational data carmot be considered complete if it
does not include dynamic methods. Few complete evaluations exist at present of the
relative and absolute topographies of the isobaric surfaces for larger oceanic regions.
The available data is in most cases insufficient for this, since it requires a reasonably
uniform network of stations over a rather extensive area. Of surveys of this type which
have been made may be mentioned : the regular series observations of the International
Ice Patrol Service near the Newfoundland Banks; those made by the "Marion" and
"General Green" expeditions in Davis Strait and the Labrador Sea by Smith, Soule
and Mosby; in the eastern North Atlantic by Helland-Hansen and Nansen; in the
Caribbean Sea and the Cayman Sea by Parr; in the Antarctic Ocean by Deacon; in the
Gulf Stream area by Iselin and Dietrich; in the area off the Californian coast by the
Scripps Oceanographic Institution La Jolla and in the area east of Japan. A complete
dynamic evaluation of the observational material accumulated for the whole of the
Atlantic is given in the "Meteor" report. The results of these surveys will be discussed
later in connection with the flow conditions in individual oceans.

The observational data for investigations of this type must satisfy certain demands.
In the first place they must be as homogeneous as possible and this can only be achieved
by a collection of the data according to uniform principles, and by a critical dynamic
evaluation using standard methods. Strictly speaking, the data should be collected
synoptically, but this cannot be done by expeditions using only a single vessel. For
larger oceanic areas it is customary, if there are no pronounced seasonal variations in
the current conditions, to combine all the available series observations and consciously
abandon the ideal of simultaneous observations. It lies in the nature of such a pro-
cedure that a representation of the phenomena in this way cannot, of course, show
individual details and the resulting charts only contain the main features. Repetition of
such surveys in the same area shows to what extent the topography remains stationary.
More importance will certainly be attached in the future to the need for simultaneous
synoptic observations. This, however, will require a greater number of oceanographic
vessels doing survey work in groups at the same time, or simultaneous recording
instruments put out into the open ocean to form a synoptic network to be collected
later.

Observational data, as well as being homogeneous and synoptic should satisfy a
further requirement which is equally not easy to fulfil. This is the density in the station
network necessary for each oceanographic survey. If only the major features of the
phenomena are required, then the interval between stations customary for oceano-
graphic expeditions (50-150 nautical miles) seems to be sufficient. Data collected on
this basis will not give refined details — neither in the distribution of the oceanographic
factors nor in topographic charts. It also will give no idea of differences between small
oceanic areas. It is, however, difficult to specify just how dense the network of stations
should be in order to obtain a representative picture of oceanographic conditions.
These questions are closely connected with changes in the oceanographic factors with
time and it is obvious that these variations cannot be studied by a single oceanographic
vessel alone. These essential questions of oceanography were dealt with in detail by

506

Ocean Currents in a Non-homogeneous Ocean

Helland-Hansen (1939) and others. The surveys made in the southern part of the
Norwegian Sea during June/July 1935 (station interval 20 nautical miles) and June/ July
1936 (station interval 10 nautical miles) have show^n strikingly large changes in the
appearance of different water types which must be due to both, to changes in time and
to local variations over short distances. In most cases they could be most probably
related to stationary, and in some cases also to progressive vortices. Such disturbances
are apparently characteristic of many more intense ocean currents in which there may
be waves and vortices of larger dimensions.

As in the Norwegian Sea, large and regular local variations, as well as variations in
time of the different oceanic factors, will also be present in the open ocean, especially
in the upper layers, and for larger distances between the stations these can introduce
an unpleasant degree of uncertainty for the dynamic preparation of the data. Only by
this can it be understood why discrepancies between the results of different investi-
gators for a particular area occur and why representations of the same oceanic region
often deviate widely from each other. Helland-Hansen has presented an instructive
schematic example showing how difficult conditions may be.

Figure 234 shows two neighbouring profiles / and // through a strong current taken at
two different times A and B. The vertical lines represent the position of the stations on

A*B

Fig. 234. To the critical discussion of a joint scientific use of observational data, which are
gained in a non-synoptical way.

which the profiles are based, which were different in both cases. In the first survey (^4)
the horizontal section at a level k was obtained directly from the vertical profiles.
The curves represent isotherms, isosteres or similar curves. Below this are shown the
conditions of the second survey {B). Thus it is assumed that the current was the same
during both surveys, but that at the time of the second survey it had been displaced
relative to the first survey somewhat to the right. In an oceanographic survey in the

Ocean Currents in a Non-homogeneous Ocean 507

open sea it is either entirely impossible, or possible only with great loss of time, to
place a station in exactly the same position as in a previous survey. The assumption in
the figure, that the stations of survey B are halfway between those of survey A, is
probably exaggerating matters a little. Since the conditions have apparently not
essentially changed, it seems to be justifiable in spite of the rather wide distance between
the stations to combine the data from both surveys as has been done below. However,
the conclusions drawn from this section are obviously erroneous. For a large station
network conditions may be the same, even in the absence of variations in time, when
deahng with a single oceanic area where large local differences are present (stationary
vortices, strong deflections of the current and others). In such cases (macro-turbulence
of the flov/) only a dense network of more or less synoptic character would then result
in a correct picture of the oceanographic conditions.

There are an additional number of sources for errors in the calculation of topo-
graphies that should be mentioned here. In the usual calculation of dynamic depths at
fixed standard pressures one proceeds according to equation (IX.9), so that the values
of the specific volume a found at certain depths (given in metres) were actually found
at depth (given in decibars). At the same time the integral values of/? are put equal to
the depths given in ordinary metres. The integral expressions for the dynamic depths
D will thus be about 1% (or at the most 2%) too small. A further error results from
the uncertainty in the a-values, especially in the upper layers, due to errors in depth in
series measurements when there is a large vertical gradient in a. Proof can be given
that an error €„ in a at a depth //„ will give rise to an error h,^ in D which can be
calculated from the equation

. _ K+i — /?»-!

where A„+i and /?„_i are the observed depths immediately above and below Z/^; if the
error at all depths is equal to e then the total error in D will be 3„ = eh, where h is
the total depth of D. In general, these errors are not large, and they can be avoided by
calculation of a second approximation but this is rarely done. Parr (1936, 1938 b) has
given an emphatic warning against uncritical use of the dynamic methods and has
pointed out that no more can be expected of these than their simple assumptions
permit. The calculations are seldom so accurate that the stream lines obtained can
be regarded as actual trajectories as should be the case for steady currents. The
stream, lines determined by the dynamic method are connected only with a single
isobaric surface and this may also give rise to erroneous conclusions. In reality they
are not subject to this constraint. Vertical displacements are also possible. This plays
probably a role in areas of upwelling water.

Another circumstance is of much greater importance. The oceanic structure at
stations where there are strong vertical density gradients depends on the occurrence
of internal waves. With these the water masses in a water column are displaced in a
periodic way and these periodic variations in oceanic structure will show in the
dynamic evaluations made for that station. The magnitude of such effects can be
judged upon at anchor stations, where repetitions of series observations at short
intervals are made. Dietrich has calculated an example of this type (Table 1 37, "Meteor"
anchor station no. 197, series 9). During the period of the measurements the physical

508

Ocean Currents in a Non-homogeneous Ocean

Table 137. Extreme positions of the isobaric surfaces at the ''Meteor'" St.
197 (8-7° S., 16-6° W.);for comparison ''Meteor" St. 198
(9-0° S., 19-8° W.) (reference-level at 3000 decibars)

Anchor St. "

Meteor" 197

Pressure
(dbar)

Difference
(dyn. cm)

"Meteor" St

Maximum

Minimum

198

position

(dyn. cm)

226

216

224

209

199

206

100

193

184

190

150

182

174

179

200

174

167

171

300

161

155

159

500

139

135

138

1000

2S00

3000

sea level showed displacements of about 10 dynamic cm and even at 1000 m the varia-
tion was still about 3-2 dynamic cm. Similar calculations have been made by Seiwell
(1932) for an "Atlantis" station to the north-west of Bermuda. These show a displace-
ment not only of the absolute position of the pressure surfaces but also the horizontal
pressure gradient is influenced. Comparison with the neighbouring "Meteor" St.
198 shows the magnitude of such variations in the pressure gradient due to the
passage of internal waves, unless these are only simulated. It is therefore not surprising
that the dynamic topographies remote from strong currents can be very confused and
only can be looked upon with utmost caution and criticism.

8. The Determination of Water Transport in Density Currents

The methods of topographical cartography of the isobaric surfaces allows an
insight, with the limitations discussed in the previous section, into the structure
of a density current, and when the pure drift and gradient currents are added,
a complete picture is obtained of the Ekman "elementar" current at any particular
place. In many cases it will scarcely be possible to give accurate details about each of
the constituents of the "elementar" current and the complete current structure will be
so complicated that it can only be shown graphically or by means of numerical
tables. Even a less detailed knowledge of the current conditions would be of con-
siderable value. Ekman (1929), by calculation of the current transport of a convection
current, first showed the possibility of obtaining information on the total mass of
water carried by a current in a relatively simple way without investigating the individual
layers. He was thereby able to show that this method of calculating the mass transport
is entirely independent of any arbitrary assumptions about friction, which seems to be

Ocean Currents in a Non-homogeneous Ocean 509

particularly valuable since this is not the case of the individual constituents of the
"elementar" current.

{a) Volume and Mass Transport

If the action of the wind at the sea surface producing a drift current almost indepen-
dent on the mass structure is disregarded, then the "elementar" current according to
Ekman (see p.41 3) is made up of a pure gradient current and a density current. The gradi-
ent current depends only on the slope of the physical sea level and, if the bottom layer is
disregarded, represents a flow in geostrophic balance independent on the depth
(equation X.4), while the density current depends only on the distribution of mass in
the interior of the ocean. This mass distribution allows to evaluate the relative vertical
velocity distribution in the density current (equations XV.7 and 12). The total vertical
pressure distribution in the ocean equally is composed of two parts. The first one
originates from the slope of the physical sea level and is independent on the depth.
If the deviation of the sea level from its equilibrium position (level surface) is denoted
by ^ (positive upwards), then the resultant pressure disturbance will be Ap = gpt,.
The second part p, originates from the mass distribution in the interior of the sea so
that p = Pi-\- Ap.

For a steady frictionless state the equations of motion will be

u = --.i^-\-U and v= -. P + K, (XV.16)

f dy f ox

where U and V are the components of the geostrophic current (equation XIII.4).
The first term on the right-hand side in these equations gives the density current which
v\'ill have velocity components

Introducing instead of pressure the dynamic depth of the isobaric surfaces D, accord-
ing to (IX. 8), and taking into consideration that the unit of the potential is —lOD
(equation IX.4), then one obtains from

10 cD , 10 8D

u, = -^ ^ and Vi = - -y -^ , (XV. 18)

f dy ' f dx '

whereby D can also be replaced by the anomaly of the dynamic depth. The volume
transport in a horizontal flow is given by

^x —

u dz and 5„ =

vdz (XV. 19)

and the mass transport by the components

M:,

■d

pu dz and My =

pv dz, (XV.20)

where d is the depth of the sea. The corresponding quantities for the pure gradient
current can be written down immediately since it is independent on the depth. The

510 Ocean Currents in a Non-homogeneous Ocean

volume transport of the density current can be determined from (XV. 1 8) , if the vertical
mass distribution is known. One obtains

S. = -^ \ -^-dz and Sy=--.\ -^ dz (XV.21)

10 f'^ dAD

Using the equation defining D (equation IX.9) a quantity

Q=\ hdpdz {XN21)

can be introduced in equations (XV.21) giving

^^ = 7 -dy ^^^ ^^^-Jd^- ^^^-^^^

Since the anomalies of the dynamic depths of the isobaric surfaces are always calcu-
lated when evaluating observational data, it is always possible to obtain the volume
transport of the density current without difficulty.

A determination of the mass transport of the density current is considerably more
difficult. The first theoretical calculations of this type were made by Ekman (1929)
who has given later (1939) a detailed and extensive account of this and of the related
problems. He obtained also formulae similar to (XV.22) but rather more difficult to
evaluate; it involves the pressures at given dynamic depths which are usually not
calculated during the dynamic preparation of observational data. Since the mass
transport can be obtained with sufficient accuracy from the volume transport by multi-
plication with the mean density, it is not necessary to calculate it independently.
Calculations of the volume transport and the introduction of the quantity Q have been
done by Jakhelin (1936). For the practical application of the equations (XV.21 and
22), p has to be taken in decibars and the depth d in metres which both can be expressed
approximately in the usual way by the same figures. This inaccuracy leads to values of
Q which, as was shown by Jakhelin, are systematically about 1 % too low. Although
the errors are small, it is nevertheless desirable to apply a correction for this to the
calculated values. The volume transport between two stations A and B [B to the right
of ^ at a distance L) is thus finally

S = L\ V dz = y

(ADA-ADB)dz. (XV.24)

It depends only on the dynamic depth anomaly at the two stations and is independent
of the distance between them. In this way it is also independent of the mass distribution
within this space. Lines of equal Q can be drawn for any larger area. Their direction
gives the direction of the volume transport and their spacing at any point is propor-
tional to the volume transport. This proportionality factor, however, depends on the
latitude. The same "current amount" does not flow everywhere between each pair of
Q-lines ; for a current towards the north and south the transport in the flow direction
will decrease and increase respectively.

For more extensive oceanic areas this dependence on the latitude cannot be neglected.
Attempts to show the changes in transport with latitude directly on a transport chart,

Ocean Currents in a Non-homogeneous Ocean

511

by drawing isolines of the quantity g/(sin 4>) instead of the ^-lines, are based on
incorrect reasoning because lines obtained in this way are then no longer stream lines ;
they will be intersected by the flow and thus lose their meaning. It is therefore better
to retain the g-lines (Thorade, 1937 b). Volume transport charts over more extended
oceanic areas have not yet been prepared, although the complete dynamical evaluation
of the observational data for such an undertaking would be available.

{b) Water Transport in Coastal Currents

Werenskiold (1935, 1937) has presented a very convenient method for the calcula-
tion of the volume transport in coastal currents, for which, in a cross-section at right
angles to the coast, a lighter water is spreading out in a wedge-form on top of a heavier
slowly moving and almost homogeneous water. Figure 235 shows a vertical section across
a current between two stations A and B. The .v-axis is placed in the sea surface in the

Fig. 235. To the computation of the water transport in a coastal current (according to

Werenskjold).

direction A-^ B and the water depth is denoted by z. In the section there are drawn
two isopycnals p and p -[- ^p and two plumb-lines x and x -f ^.v. The boundary
surface of the wedge-shaped top layer forms the isopycnal pi reaching the surface at
C. The top-layer has a depth z^ at point x, however, the depth Z^ at the station A.
At an arbitrary point M on one of the plumb-lines (density p) the component of the
velocity of the density current at right angles to the section will given by the equation
(VII.8):

Vi =

fp.

i dp-

Thereby j is the slope of the isopycnal which is dependent on .y and z. Denoting
Sl fPm by b, then one obtains from the relation above

-bj

dz
dx'

(XV.25)

where the derivative dzjdx has to be taken along an isopycnal, that is, for a constant
p. The volume transport at a plumb-line can then be obtained by integration from
0 to r^. By partial integration one obtains

S - Vi dz

Vi-

zdVi.

512 Ocean Currents in a Non-homogeneous Ocean

The first expression on the right-hand side is zero, since fj = 0 for z^ and thus using
(XV.25) one obtains

Pi dz ^ b [Pi dz^ ^

Po (l-"^ 2 \o.dx

Pq is the density at the sea surface. The total volume transport through the entire
top layer from C to station A is thus finally obtained by integration from x,. to xa

XA h

Sdx = ^

Pi dz^ ,

~r-dp.

The integral of (dz'^jdx) dx is equal to Z^ where Z is the depth of the isopycnal at the
station A. Finally, on repeating the partial integration, since Z is zero at the sea surface,
we have

St-2f

'-^^ Pi — P

dZ\ (XV.26)

If the transport between two arbitrary verticals A and B is required, then the expres-
sions (XV.26) are evaluated at both places and the difference is taken. The water
transports obtained in this way are subject to the same limitations for the quantity^.
It is noticeable that a knowledge of the mass structure at the two stations is sufficient
for the determination of the transport through the vertical section between them,
without having a knowledge of the distance between the two stations. Werenskiold
offered an explanation for this fact by pointing out that the flux in horizontal direction
through the section is unaffected by stretching or shrinking of part of this section,
because the pressure gradient and therefore also the current intensity are changed
inversely proportional to the current width, and the distance between the two stations
is eliminated. It seems, therefore, that only the mass distribution of a single station is
required in order to calculate the transport through a vertical section by means of
equations (XV.26). However, this is not true at all since a knowledge of the stratifi-
cation at two stations C and A is required and, furthermore, the water at C is homo-
geneous and has the same density as the deep water at A.

Since the integration of equation (XV.26) is performed using ordinary metres, the
correction required previously for Q is not needed here.

Chapter XVI

Currents in a Strait

1. Water Stratification and Water Movements in Sea Straits

Sea straits connect the open ocean with mediterranean and adjacent seas. By means of
the water flux through the connecting straits directed towards the open ocean a medi-
terranean sea can often produce considerable effects on the oceanographic conditions
in the open ocean. This influence is sometimes so powerful as to involve entire parts
of an ocean, changing drastically the oceanic conditions in these parts. Present
knowledge of oceanographic conditions in sea straits is only partly satisfactory. The
main outlines and the typical features are known but much remains to be explained
especially in the details, that will require systematically arranged observations and
measurements.

The continuous interchange of water between mediterranean and adjacent seas
which are completely surrounded by land and the open ocean is controlled very
largely by two factors :

(1) by the proportion between fresh- water inflow (precipitation and run off (river
water and other water)) and evaporation in the mediterranean sea, and

(2) by the depth and width of the passage to the open ocean, that is, the morphology
of the sea strait.

The currents in a sea strait are a consequence of the difference in vertical thermo-
haline stratification between the water masses in the adjacent sea and that of the open
ocean off the entrance to the strait.

Sea straits can be divided on the basis of the currents flowing in them into two
groups :

(1) Those in which the adjacent sea is surrounded by arid land masses. Here
evaporation exceeds precipitation (E — P) > 0. The loss of water due to this excess
must be replaced from the open ocean through the strait.

(2) If the entire oceanic area lies in a humid climate (E — P < 0), then the excess
of precipitation over run-off will flow out into the ocean through the connecting
strait.

To the first group belong — inside the area of the Eastern Hemisphere rich in
evaporation and with little precipitation — the Strait of Gibraltar, connecting the
Atlantic with the high-salinity European Mediterranean; the Strait of Bab el Mandeb,
connecting the Indian Ocean (Gulf of Aden) with the highly saline Red Sea and the
Strait of Hormuz between the Arabian Sea (Gulf of Oman) and the Persian Gulf,

To the second group belong — in the northern humid region — ^the weakly saline
Baltic Sea which is connected by way o^tiarrow belts and Sounds through the Kattegat
and the shelf-like North Sea with the open ocean; the predominantly humid Black
Sea connected with the arid Mediterranean through the Bosporus and the Dardanelles;

513

514

Currents in a Strait

the White Sea and the Barents Sea with the so-called Gorlo and the Gulf of St Lawrence
connected with the Atlantic by the Cabot Strait and others.

The interchange currents in all these sea straits occur characteristically on two
different levels; there are always two currents in the strait, one above the other. The
upper layer always flows toward the sea having greater density, the lower layer in the
opposite direction, and between them there is usually a well-developed discontinuity
layer in the density field (see Pt. I, pp. 133 and 182-184 (Figs. 56, 83-85) on the general
distribution of temperature and salinity in sea straits). Thus in straits of moderate
width there are always two water bodies one above the other with a boundary layer
between them sloping down from the sea with the greater density towards that with
the lesser. The wedge-form of these superimposed water layers along the strait is a
characteristic feature of the structure of the water masses in a sea strait. Table 1 38 gives
a summary of mean density in the upper and lower water layer and of the slope of the
boundary layer for some sea straits in different climatic regions (Vercelli, 1929;
MoLLER, 1931). The greater the slope the smaller the density difference, i.e., the
slower the interchange movements. In addition to this effect of the density differences
other circumstances also control the slope of the boundary layer, particularly the
bottom topography of the sea strait, because it affects the continuity requirement of a
complete balance between the mass transport in the upper and lower current under
stationary conditions. For example, in the Bosphorus, the slope of the boundary layer
is strongly dependent on the bottom inclination and because of this the wedge-form
character of the lower water is lost there.

Table 138. Mean slopes of the boundary layer and mean densities of the
upper and lower water in several sea straits

Sea strait

Mean
width
(km)

Mean
length
(km)

Minimum

depth

(m)

Boundary

layer slope

(m/km)

Mean

density

Difference

Upper
water

Lower

water

Danish sounds (Belts)
Dardanelles .
Bosphorus

Gibraltar
Bab el Mandeb

ca.lQ

4-5
0-7

ca. 100
60
30

60
134

6-9

57
37

333
185

012
0-20
M3

4-2
3 0

13-5
180
13-5

26-8
259

23-5
28-8
27-5

28-8
27-4

100
10-8
140

20
1-5

Besides this longitudinal slope there should also be a transverse slope of the boundary
layer due to the effect of the Coriolis force. The faster the currents and the wider the
strait the greater will this slope be. If the upper homogeneous water mass in the
strait has a velocity u^ and the lower one a velocity lu, and if the transverse inclination
of the sea surface is given by 8l,^jdy and that of the boundary surface between the upper
and the lower current by ^i^i^y^ then, under stationary conditions the equations

g-^=-fih and
will be valid where (/= 2aj sin </•).

PoUo — piUi
P2 — Pi

(XVI. 1)

Currents in a Strait 515

In Fig. 236 which shows a cross-section through the strait, u^ (upper current) is
positive and Mg (lower current) is negative, and as a consequence the sea surface rises
to the right while the boundary surface between the two water types slopes downward
to the right. This latter inclination is considerably steeper than the first. For a certain
definite velocity ii^ the water mass of the upper current may be too small to cover,

JIv

Fig. 236. A model in order to study the thermo-haline circulation in sea straits.

during its displacement to the right, the whole of the lower water mass. Choosing in
equation (XVI. 1)

^ = 4^ = - ^«i, (XVI.2)

/= 10-^ g = 1000 and u^ = 100 cm/sec, then Ah = 10-^ X L, where Ah is the
elevation of the water level for a given width L of the strait. For the Dardanelles
L = 5 km and therefore Ah is 5 cm; in the Strait of Gibraltar L = 20 km and Ah is
20 cm. The latter value is already quite large. For quite a large width of the strait the
inclination may be so steep that in a narrow strip along the coast the opposite moving
heavier water may rise to the surface, so that in the strait at the sea surface there will
be a front with currents flowing in opposite directions on either side. In narrower
straits transverse slopes of this sort will be barely detectable.

The internal structure of both water bodies is usually stratified; however, this
stratification is only slight in salinity, but at time it may be pronounced in the tempera-
ture. In all cases in low-salinity seas where the access depth to the strait is deeper than
the discontinuity layer, due to increased radiation in summer, a temperature inversion
will be formed within the upper current with a minimum above the boundary surface ;
below this in the lower water a secondary maximum appears and then the temperature
will decrease again to the bottom value. Figure 237 shows two vertical temperature and
salinity curves of this type for the northern parts of the Bosporus. The temperature
minima always decrease in the direction of the surface current due to the effect of
mixing.

The discontinuity layer in the salinity remains fairly sharp along the total length of
the strait, though in each of the water bodies the absolute values will change somewhat
due to mixing: in the upper water body the salinity will therefore usually increase and
in the lower it will decrease. The changes in the Bosphorus and the Dardanelles are
thus over 300 km about 10%o. Similar values have been found in the Danish sounds
(Belts). The greatest changes are, of course, usually found where there are large
irregularities in the bottom topography where eddies and vortices are generated.

As a further characteristic phenomenon found in sea straits the boundary layer
between the water bodies often does not coincide with the level of reversal of the

516

Currents in a Strait

current direction (level of no-horizontal motion). The latter surface can be found
either above or below the boundary layer between the two water bodies, but the
height-difference is never large. This phenomenon has been observed in the Bosphorus,
the Dardanelles and in the Strait of Gibraltar. As has been shown theoretically
(NoMiTSU, 1927) the two layers can coincide only when the water bodies are com-
pletely homogeneous. Deviations from such a state are due to mixing processes
occurring at the boundary surface between the two oppositely moving currents,

/, "C S, %o i^, cm/^ec

10° 12° 14° 16 24 32 40 20 ^ 60 100 140

t'"-

i 1

1 1

~-«.^

Fig. 237. Vertical curves of temperature, of salinity and of the current for always a single
station in the northern and southern Bosporus. Full lines: station 110 in the section
Karibdian Burnu-Porias Burnu, 12 May 1918. Dashed lines: station 123 in the section Orta
Koi-Istawros, 23 May 1918. At the current curves the arrows in the current are situated
such that: west, towards left; east, towards right.

The currents within a strait are completely known only in the Bosphorus and in
the Dardanelles where accurate current profiles have been obtained by Merz (Moller,
1928). For other sea straits current measurements have been made for short time
intervals only and at few stations. In general, the greatest velocity in the upper current
is found close to the sea surface. In a cross-section the current distribution corresponds
to that of a river in which the lines of equal velocity (isotachs) follow approximately
the river bed. Due to the wedge-form of the upper water body the velocity increases
in the direction of the current (in the Bosphorus and in the Dardanelles from 50 to
150 cm/sec). The transition from the upper to the lov/er current does not occur dis-
continuously but increases in sharpness as the density jump becomes greater. This
phenomenon is also due to the turbulence of the current, which is strongly suppressed
when there is a large density discontinuity in the vertical. When the depth of the water
is not too great the lower current follows the bottom topography of the sea strait,
and therefore the cores of the upper and the lower current need not lie exactly above
each other. The vertical current distribution in the lower current shows a maximum in
its central core situated about half way between the current reversal layer and the
bottom. For continuity reasons a decrease in velocity occurs in the lower current if
the depth increases along the strait; however, if the depth decreases there will be a
corresponding increase in velocity. At the bottom the velocity may be so intense that
it causes considerable erosion. The occurrence of rolls in a depression of the sea bottom
(Koike, deep hole) may simulate a decrease of the velocity to zero near the bottom
(see p. 390).

Currents in a Strait

517

A special phenomenon found in the current systems in irregular shaped straits is
the occurrence of stationary vortices with vertical axes. They are used with considerable
advantage in navigation. In the Bosphorus and in the Dardanelles they are well
developed in both surface and bottom currents (Fig. 261). In some cases two stationary
vortices are formed side by side with an opposite sense of rotation so that side branches
("neer" currents) develop returning later into the direction of the main current.

Table 142 gives an idea of the very large amounts of water passing through major
sea straits. In broad and deep straits, such as the Straits of Gibraltar and Bab el
Mandeb, the transport may be 5 to 20 times greater than in narrow shallow straits.
Expressing the amount of inflow or outflow by means of water-level change (in mm)
of the total Mediterranean, a particularly clear idea of the great difference between
the humid and the arid, semi-arid areas, respectively, is obtained.

2. Theory of Currents in Sea Straits

The dynamic cause of currents in sea straits lies in the density difference between the
open ocean and the enclosed sea, or more exactly, in the density difference at the
level of the bottom of the strait between the entrance to the strait and its outlet. The
thermodynamic mechanism inherent in this circulation can be demonstrated by a
simple model (Defant, 1955) (Fig. 238). In a strait with a depth h limited at ad and

-i-h-

FiG. 238. Cross-section through a rectangular sea strait and its current system.

be at either end by two water columns belonging respectively to the ocean and the
enclosed sea, the system will be made up of horizontal layers of water 1 and 3 between
ah and cd, respectively, and vertical columns 2 and 4 between be and ad, respectively.
We assume a stationary state and neglect — because of the narrow strait — the effect
of the Coriolis force. The currents which adjust under stationary conditions must be

Table 139. Water transport through sea straits {according to Moller)
( + , current from the adjacent sea; — , current into the adjacent sea)

Sea strait

Upper

current

(km^/year)

Lower

current

(km^/year)

Outflow

amount

(km' /year)

Outflow
height
(mm)

Adjacent sea

Area
(10' km2)

Danish sounds (Belts)

Bosphorus

Dardanelles

Gibraltar
Bab el Mandeb

+304
+398
+ 591

-55198
-16450

-152
-193
-386

+ 51886
+ 12800

+ 152
+205
+205

-3312
-3650

+ 383
1+488

-1330
-7980

Baltic Sea
Black Sea

Europ. Medit.
Red Sea

397
420

2496
458

518 Currents in a Strait

solely antitryptic, that is, the pressure and the frictional forces will be in equilibrium
see p. 323). The system will be subject to the equations

1 dp
Y - Ri = 0 0= 1 and 3) along 1 and 3 (XVI.3)

and

1 dp
g ^ - Ri = 0 (/ = 2 and 4) along 2 and 4,

where i?, is the effect of friction along each of the sections. Multiplying the equations
(XVI.3) by p and integrating along the individual sections one obtains, after adding,
the relations:

g( \dz-\-\pciz~(()pR,ds] = 0, (XVI.4)

\ Jb Jd J abed I

g{P2 - PA)h = (b pRi ds. (XVI.5)

J abed

Here pg ^^id p^ are the mean densities in the vertical water columns 2 and 4. All the
quantities Ri are positive and depend on the current velocity. From (XVI.5) it follows
then that the left-hand side must also be positive. That is, the mean density in be
must be greater than in ab or p, > Pi- This relation thus fixes the direction of the
current in the strait and also give the dependence of the current velocity on the
density distribution in the water masses. This can be used to find an approximate
value for the current velocity maintained by the thermodynamic forces acting inside
the system. According to the circulation theorem, when a = Xjp

- i adp = i Rids XVI.6)

J abed J abed

and since Z),

a dp

gives the dynamic depth of the pressure surface p in the water column /, we obtain
from (XVI.6)

D,- D, = (/?! + i?3)/ + (^2 + ^4)/^. (XVI.7)

The integral — j adp is the work done by the pressure forces in the system ; if it is
positive, this work can thus be balanced by the work required by the friction. The
relation (XVI.6) states that, in the thermodynamic machine the expansion takes place
at a higher pressure than the contraction. Since an expansion is associated with an
input of heat and a contraction is associated with a heat loss, the heat gain must
therefore occur at a higher pressure than the heat loss. Actually, in the model of
the sea strait in point there will be a higher pressure and a higher temperature, the
latter due to a greater heat gain. Such a sea strait system is thus a true thermodynamic
machine in action.

The current intensities in a strait can be calculated approximately by means of the
above equation (XVI.5). For a channel of length /, if friction is neglected in the vertical
part of the circulation, the equation will take the form

2pR/ = g{p, - p,)h. (XVI.8)

Currents in a Strait 519

In addition, it is necessary to make an assumption about the dependence of the
friction on the current velocity. For a shallow current it is possitble to put R equal
to Kpu^ dyn/cm^ (see equation X.9). However, for each horizontal branch

and the friction per unit mass of this branch is

The total friction is therefore given by

/c(2m)2

and the equation (XVI. 8) thus gives an equation for the determination of the mean
velocity in one water body

6 Kl p

If the dimensions of the strait are known, w can be calculated. Only the value of the
Taylor frictional constant requires a little further comment. For a smooth channel
K has been found experimentally to be 0-0025. It cannot be expected that the value of
K will be as small as this because of the irregular configuration of the sea bottom and
sides of an actual sea strait. In rivers, for example, k may be as much as 10 times this
value or about 0-03. Considerably higher values of the boundary friction are there-
fore to be expected due to the roughness of the bottom in a somewhat wider strait.
A proof of this is the frequently observed sharp decrease in velocity in the layer next
to the bottom.

Choosing mean values for the dimension of a sea strait, for example, / = 50 km,
h = 100 m and the difference in density/!/) = 10 x 10"^, according to Table 140, then
putting K = 0-03 the equation gives w = 28 cm/sec which accords with the average
velocities found by observation. In the Danish sounds (Belts) the velocity of the current
is about 10 cm/sec, in the Dardanelles about 25 cm/sec, in the Bosphorus 30 cm/sec,
in the Strait of Gibraltar 30-35 cm/sec and in the Strait of Bab el Mandeb about
40 cm/sec. The calculated value fits thus very well in this series of observed values.

For a detailed theory of currents in sea straits it is necessary in the treatment of the
stationary state to return to the antitriptic equations of motion in which the gradient
force and all the frictional forces are always in equihbrium (Defant, 1930). A suitable
model is a rectangular channel, depth h^ and length L, connecting two seas with differ-
ent thermo-haline structures. Both water types are homogeneous (upper water density
Pi, thickness in the middle of the channel h^ ; lower water density pa^ thickness in the
middle of the channel h^ — fh over a plane bottom). The co-ordinate origin is placed
in the middle of the channel at sea level with the positive r-axis directed upwards.
The upper current flows in the direction of the negative .v-axis (see Fig. 239) and the
physical sea level must therefore also slope downwards in this direction (pure slope
current). The static pressure in the upper layer [z from l,^ to — {h^ — Q] will be

520

Currents in a Strait

Fig. 239. To the theory of ocean currents in sea straits.

p^= p^-\- gpi(^i — z), however, in the lower layer [z from — {h-^ — i^to — //g] will be
P2= Po-\- S(p2 — Pi)(^2 — fh) + ^Pi^i — gp2=- Po is the atmospheric pressure at the
sea surface. Putting /Jq = — gPiC then ^ is the displacement of the sea surface produced
by an atmospheric pressure p^. The equations of motion in the stationary state,
disregarding the Coriolis force and friction on the sides of the channel are then

-^8-x^^^

0 +

7] 8^Ui

(XVI.ll)

Pi ^ /y y\

Pi ^^2

8^Uo

+ - — - = 0

P2 Sx p dz^

If Ci and ^2 are small compared with the depth of the strait then, for a linear slope of
the physical sea level, u^ and Wg will be independent of x and the continuity equation
will take the simple form

- /,, - hi

i^dz^O. (XVI. 12)

Ml dz + U2

Jo J -hi

The volume transport of the upper current must be equal to that of the lower current.
The boundary conditions are as follows :

( 1 ) If there is no wind, dujdz = 0 when z = 0. The effect of a wind along the channel
can be taken into account by the assumption

8ui

^1 Pa W'-,

where Pa is the density of the air, k^ is the Taylor constant (equation X.9) and n- is the
wind velocity relative to that of the water. Taking diijdz = M for z = 0 allows the
effect of the wind to be taken into account.

(2) At the boundary surface there will be a reversal of the current direction, that is,
when z = — hi, then Ui = U2 = 0 (no horizontal motion).

(3) At the bottom (z = — h^ three different cases of boundary friction can be

Currents in a Strait

521

considered : adhesion to the bottom u^ = 0, ghding du^jQz = 0 and average frictional
influence r](8u2ldz) = Kp^ ul. If the roughness of the sea bottom is shght the factor k
is of the same magnitude as k^ ; for a rough bottom it has been found in hydrauhcs
to be about 10 times greater.

Solutions of equation (XVI. 11) can be given for all three cases. For the extreme
cases of adhering (haften) and gliding (gleiten) and with uniform atmospheric pressure
(I = 0) one obtains

Slope of the physical sea level :

Slope of the boundary layer: /g = 7-

Velocity of the upper layer :
Velocity of the lower layer :

aiz^ - hi) + M{z + /?i).

.(XVI. 13)

adhering :
gliding
where

u., = A(z + h^(z + fh) with A =

(i-)

U2 = A [(z2 - hi) + Ih^iz + h,)] with A =

4[\

l-^A

Pi .

and m = 4a

Because A is always negative, the slope of the internal boundary surface will always
be opposite to that of the sea surface ; however, because of the density difference
(pa — Pi) in the denominator it is always considerably larger. The slope of the boundary
surface found by observation is a function of the water interchange between the two
seas. The currents in the two water bodies always flow in opposite directions. The
current profile in both water bodies is of a parabolic form. In the upper current the
maximum occurs at the sea surface; if the wind is in the direction of the upper current
it will decrease rapidly with depth, but if the wind is against the upper current the
decrease will be small. The upper water in this case will be piled up against the current.
If there is a very strong wind at the surface against the upper current, the current
maximum may be somewhat below the sea surface. All these theoretical conclusions
are in complete agreement with observation. In the lower current the velocity maximum
will adjust in variable depth below the boundary surface according to the variable
friction at the sea bottom. If there is adhesion it will appear in the middle part of the
lower layer, if there is gliding at the bottom it will occur at the bottom itself and for
moderate friction it will be situated between the discontinuity surface and sea bottom.

Numerical values corresponding roughly to those for the Bosphorus may be taken
as an example: length of the strait = 30 km, depth = 70 m; upper layer p^ = 1-013
down to 40 m; lower layer p^ = 1-027, p2 — pi= 14 x 10"^; slope of the physical
sea level 6 cm in 30 km, -qj p = 250 cm^/sec, which is about the same as the frictional
coefficients for tidal currents; wind = 5 m/sec along the strait. For the slope of the
boundary surface (metres in 30 km) the equation gives the values contained in Table
142.

522

Currents in a Strait
Table 140. Slope of the boundary layer {given in m/30 km)

For adhering

For gliding

Moderate friction

Wind with the upper current
Wind against the upper current

44
53

14
17

33
37

The magnitude of these values is similar to those actually found in the Bosphorus
which average about 34 m. In the case of a south-west wind the slope is steeper than
for a north-east wind, which agrees with the theoretical result. Figure 240 shows the

V, cm/sec
40 60

100

£40

a>
O

■ I

— ' 1

■■ 1

^.i-:;

**»gj

^^^

~'-^=-'=r^

_^^

mmmm

,.-' \

V, cm/sec
0 20 40 60 80 100

E
i 40

-i —

)

■ 1

— 1 —

'^■Mfa-:

"^^^

"^^

?^- ^

y \

^^^

Fig. 240. Vertical current distribution in the upper and lower current for a certain wind

direction at the sea surface of the sea strait (in the lower current: , in the case of

clinging to (Haften); , in the case of gliding (Gleiten); , for a medium

friction of the water at the sea bottom).

current profile in the upper and lower current (omitting signs). These values are also
in agreement in all cases with those observed in the Bosphorus and the wind effect
was also of a similar kind.

The theory is based on two water bodies that are homogeneous over a cross-
section at right angles to the strait. In nature they will be stratified and the cross-sectional
area can vary considerably along the length of the strait. Furthermore, mixing at the
internal boundary surface will tend to spread the discontinuity surface into a density
transition layer. The current boundary surface will then no longer coincide with the
lower limit of the upper water since there is no longer any sharp boundary. Then
conditions become so complex that they can no longer be handled mathematically.

Currents in a Strait 523

However, the stratification does not appear to be of decisive importance to the prin-
cipal phenomena of the water interchange and therefore the simple case of two
homogeneous water types gives the essential outlines.

3. Ocean Currents in Individual Sea Straits

{d) Bosphorus and Dardanelles

Due to the investigations of Merz and Moller (1921, 1938, with Atlas) these are
the straits in which conditions are best known. Systematic surveys along cross-sections
and longitudinal sections have given a good understanding of the three-dimensional
thermo-haline structure of the water masses and the corresponding currents in both
straits and some insight into the detailed mechanism of the processes involved. Over
the whole area of water interchange between the Aegean and the Black Sea there is a
characteristic stratification with a sharp density transition layer. From a depth of
200-1 50 m in the Black Sea it rises at the entrance into the Bosphorus to less than 1 50 m
and in the narrow part it rises rapidly to 20-15 m at Istanbul. It remains at this depth
throughout the Marmara Sea until it rises again in the Dardanelles, at first very
slowly, then more rapidly in the straits between Nagara and Tschanak to 10 m.
At the southern entrance to the Dardanelles it reaches almost to the surface. Figure 241
presents the density distribution in two longitudinal sections along both straits.

The wedge-form of the upper water shows clearly in both straits ; in the lower water
it is present only in the Dardanelles, since the sea bed in the Bosphorus slopes down-
wards towards north as much as the internal boundary surface. At the entrance to the
Bosphorus the salinity of the upper water is 16-18%o and at the outlet from the Dar-
danelles into the Aegean it is 26-28%o. Of this increase 2%o occurs in the Bosphorus,
5%o in the Sea of Marmara and 3%o in the Dardanelles. Mixing in the straits thus can-
not be very effective ; this is also shown by the maintenance of the temperature in-
version which is still partly present in the Dardanelles (see Fig. 237).

The upper current runs through the channels as a narrow band within limits set by
the projections of the coast. In several coastal bays on both sides of the straits numerous
standing vortices occur. The current profile shows that the velocity is greatest at the
sea surface and decreases rapidly with depth. Due to the wedge-form of the current it
increases from north to south; under average conditions it is 40-50 cm/sec at the
entrance to the straits and 1 50 cm/sec or more at the other end.

The lower current follows the windings of the channel more closely than the upper
current and the stream lines of the two currents are therefore not always super-
imposed. The lower current is strongest in the central parts of the lower water (in the
Bosphorus about 16 m and in the Dardanelles about 45 m above the bottom). The
velocity is 100-150 cm/sec in the Bosphorus and decreases from 25 to 10 cm/sec in the
Dardanelles.

In the straits the boundary surfaces between different currents and between different
water types do not coincide ; the first rises from north to south more slowly than the
thermo-haline transition layer and they intersect at the narrowest part of the straits.
Thus in the northern part of both straits upper water flows with the lower current and
in the southern parts lower water returns with the upper current. The changes in the
currents due to variations in wind and atmospheric pressure are pronounced. During
strong north-east wind the surface current is accelerated, the current core thereby

524

Currents in a Strait

Fig. 241. Longitudinal section of the density at. Upper picture: through the Bosporus in
Sept/Oct. 1917. Lower picture: through the Dardanelles June/July 1918 (according to

Moller).

narrows and the standing vortices increase in extent. During south-west winds the
surface current becomes weaker and broader and the lower current is accelerated.
Figure 242 gives a longitudinal section through both straits showing the currents during
a period with stronger north-east winds with a large pressure gradient towards the
south-west. This wind influence produces a strong asymmetry in the current structure.
For a period with a south-west wind the current conditions are aflTected in the opposite
way. These flow conditions, however, no longer represent a stationary state.

Currents in a Strait

525

A comparison with the theory presented above can only be achieved by means of
current profiles in which the varying effects of changes in pressure and wind are
eliminated. A computation of average profiles out of three typical ones for each
strait allows a qualitative comparison. Figure 243 shows that excellent agreement can be
obtained by suitable choice of the frictional coefficients. A numerical evaluation of
equations (XVI. 13) is given in Table 142. The average decline of the physical sea level
along the Bosphorus is about 6 cm in 30 km and is greater at the northern end, less at
the southern end. Along the 65 km Dardanelles it is only 7 cm; the value of 12 cm in

170 167 161 153 148
Sto).82 .77 69 68 63 61 57 51

180

48 45

29 23 17 3

14 35

100

Stat.
75 63 59 30 i5o56 50 39

35 23/27 19

Fig. 242. Longitudinal section of the current velocities (cm/sec). Upper picture: Bosporus
for N.E. 5 and ^p = 4-5 mm. Lower picture: Dardanelles for N.E. to E. 3^ and /!/> = 3 mm.

526

Currents in a Strait

V, cm /sec

20 40 60 80

V, cm/sec
0 20 40

. 40

Fig. 243. Vertical current distribution in the northern part of Bosporus (to the left) and of

the Dardanelles (to the right); H \ 1 , according to the observations;

. — . — • — , according to the theory.

the middle of the strait must be due to piling-up of water in the narrowest part of the
strait.

Table 141. Sea surface and slope of the internal boundary surface, as well

as frictional coejficients in the Bosphorus and the Dardanelles,

calculated from current profiles

Wind conditions

{Sea surface
boundary
surface .

Turbulent coeflRcient

(cm^/'sec)

Bottom friction k .

Bosphorus

Northern
part

NE-SW
101

298
0017

Middle
part

NE.j
5-6

371
0-155

Southern
part

NE and SSW
2-4 cm/30 km

36 m/30 km

485
0 015

Dardanelles

Northern
part

NE.3.
7-6

82
0109

Middle
part

12-2

28
0038

Southern
part

NE.3_,
7-2 cm/65 km

19 m/65 km

420
0-388

The turbulent coefficient is of the same order of magnitude as in tidal currents.
The coefficient of bottom friction has a mean value of 0-12 which is very large. The
individual values are strongly scattered but are around 50 times larger than the values
found for natural channels and about 5 times larger than those found for rivers. The
rolls with horizontal axis produced by the very irregular bottom and which cannot be
observed by means of current measurements may simulate a bottom friction larger
than actually present.

(h) Water Interchange Between North Sea and Baltic

This takes place in the area between the Kattegat in the north and the Darsser and
Drogden ridges in the south. These give access to the Baltic at depths of 18 and 7 m,
respectively. The annual inflow of fresh water into the Baltic averages about 500 km^
of which 467 km^ is the inflow from rivers and 206 — 1 82 = 24 km^ is the excess of

Currents in a Strait 527

precipitation over evaporation (Witting, 1918). This inflow of fresh water disturbs
the equilibrium between the North Sea and the Baltic and gives rise to a water inter-
change with an upper current flowing towards the North Sea and a lower current flowing
into the Baltic. Knudsen's relations (see Chap. XII. 5) aff'ord an estimate of the water
interchange balance. It appears from this that the inflow due to the lower current over
the rise on the west side of the Arkona basin is equal to the inflow of fresh water into
the Baltic and that the outflow in the upper current is twice as great. Detailed data
indicate that the decrease in the water amount being carried by the lower current
between the Skagerrak and the Baltic is opposed by a corresponding increase in water
amount carried by the upper current. Therefore important mixing processes must
always act within the sea straits.

Calculation of the proportion of water with a salinity of 33''/oo '" ^^^ lower current (see Table
above) shows that until the Fomas section, not less than 67% of the water entering the Kattegat
has mixed the water of the upper current and that almost ?0% of the remainder mixes with the upper
water before reaching the Arkona basin. Thus only 1° „ of the water of 33°/oo salinity entering the
Kattegat in the lower current finally enters the Baltic. The remaining 93 % mix with the upper water
and return to the Skagerrak. In the same way a large part of the upper water mixes with the lower
current and is carried again towards the Baltic. About a third of the water leaving the Baltic in the
upper current w est of the Arkona basin returns to the Baltic and not less than two-thirds of the water
in the under current flowing into the Baltic over the rises has come from the Baltic itself, and only
one-third is the water witha salinity of 33700 that flows into the Kattegat in the lower current (Schulz,
1930).

This applies only for the annual means. For the investigation of the water interchange in individual
months the assumption of a constant water amount in the Baltic is no longer valid, since the water
level shows an annual variation and other shorter oscillations. In some months the outflow from the
Baltic is stronger and in others less. Investigations by Witting for the period 1 898 to 1912 indicate that
there are pronounced maxima in fresh-water outflow from February to June, as well as in September.
A detailed treatment of the data on currents in the Oresund and the Belts recorded between 1910 and
1916 by Danish and Swedish light-ships has been made by JACOBSEN(1925),who foundforthe period in
question good agreement with the annual variation in water outflow from the Baltic found by Witting.

In water interchange processes two phenomena must be distinguished. The first is
the orderly steady water interchange that takes place in a strait connecting two seas of
different thermo-haline structure. This interchange is associated with the two currents
which are essentially antitryptic flowing along an inclined boundary surface. In addition
to this continuous steady water interchange there is a second phenomenon, the total
displacement in both directions of the entire water mass of the strait by the wind or
due to differences in atmospheric pressure. In the Bosphorus and the Dardanelles
these meteorological influences are of minor importance in comparison with the regular
thermo-haline water equalization, but in the connecting straits between the Baltic and
the North Sea conditions are reversed. Here the piling up of water by the wind
(" Windstau ") and by atmospheric pressure differences is so strong that the regular
steady interchange currents are almost completely masked. The main phenom^enon is
thus an irregularly occurring, occasional transport of the whole water mass in its
total vertical extent more or less in the same direction in spite of its pronounced
vertical stratification. The regular steady interchange can only be obtained by elimina-
tion of these irregular movements which can be achieved by taking mean values over
long periods. Strong tidal effects are also present and must be eliminated by a harmonic
analysis. Mean values have been calculated by Jacobsen (1909, 1912, 1913), and the
mean structure at four different stations is shown in Table 144.

528 Currents in a Strait

Table 142. Mean currents in the Oresund and in the Great Belt (cm/sec)
( + , directed towards the North Sea; — , directed towards the Baltic Sea)

Lightship

Depth

(m)

Lappegrunden

Drodgen

Schultz's Grund

Southern
Great Belt

+ 37

2-5

+47-1

+ 12-6

+2-4

—

+ 31-4

+ 110

+0-4

+ 30

-7-5

+ 8-8 (7 m)

-9-4

+ 12

-11-4

—

-18-2

-2

-7-6

—

-190

-15

-40 (23 m)

—

-150

-13

—

-13

—

-9

The lightship "Lappegrunden" hes in the most northern part of the sound, the
lightship "Drogden" lies in the central part and the Schultz Grund lightship lies in
the southern part of the Kattegat at the entrance to the Great Belt. For larger depth
the mean upper current is directed towards the North Sea and the lower current towards
the Baltic. The current profile corresponds rather well to that deduced theoretically.
In the shallow waters of the Oresund ("Drogden") the entire current from the surface
down to the bottom is directed towards the North Sea. As shown by Jacobsen, the
internal field of force is very weak here and the great width of the channel permits
cross-circulations to play an important part.

This steady water interchange produced by the internal field offeree is [superimposed
on the strong currents] almost always present in this area; which are produced by
differences in level between the Kattegat and the southern part of the Baltic due to the
piling up of water by the wind and due to differences in atmospheric pressure. These
are also antitriptic slope currents and give rise to displacements of the internal front
between the water bodies which here are situated side by side (Skagerrak and Baltic
water) (see Pt. I, p. 182, Fig. 85). Wattenberg (1941) has made a detailed investigation
dynamics of the displacement of these fronts and of the duration of the movements,
and has given a basis for the estimation of mixing in the Belts and of the flow of North
Sea water into the Baltic. There is a close correlation between the flow through the
Great Belt (computed by means of lightship current measurements) and the changes in
salinity. A rather close connection exists also with the meridional pressure gradient.
The duration of inflow and outflow periods changes, of course, according to the varia-
bility in the all-over weather situation over wide limits; the long-period variations are
well shown by cyclic variations in the salinity, since the inertia of the water masses
weakens or even completely suppresses the smaller shorter-period disturbances.
Extreme positions of the internal fronts are due to prolonged inflow and outflow
periods (Fig. 244). In the north the front may reach out from the Belt Sea as far as
into the middle of the Kattegat. Towards the south under reversed conditions the
front may in extreme cases reach the Darsser and the Drogden rises separating the
Baltic from the Danish sounds. This difference in behaviour to the north and to the

Currents in a Strait 529

south is due to the following facts. During outflow the upper water is not subject to
any resistance and may therefore spread out arbitrarily at the surface, while during
inflow the more saline water advances towards lighter water in front of it, and in this
case the bottom topography exerts great influence. On passing the rises in the south
the denser water sinks down to the bottom and the position of the front at the surface
remains fixed near the rise. In this way large amounts of highly saline water flow into
the basin of the Baltic thus renewing the stagnating deep and bottom water. Such
processes are necessarily connected with long periods of weather favourable for inflow,
which cause the front to remain in extreme southern position.

More recently, Knudsen (Jacobsen, 1936) has organized detailed hydrographic
investigations in the area to the south of Denmark. This work has been devoted mainly
to the collection of accurate records for the sections between Gedser and Dars and
across the Fehmarn Belt, thus providing continuous surveillance of the water inter-
change between the Baltic and the Kattegat.

In the strait of Gibraltar, instead of a single bottom rise there are three, all west of
Cape Tarifa. The first one extends in an arc from the Cabezos reef to Punta al Boassa
(maximum depth 320 m), the second one runs from Cape Trafalgar over "The Ridge"
(in places only 55 m deep) to Cape Spartel (maximum depth 366 m) and the third
one lies about 10-20 km west of the second with a maximum depth of over 300 m.
The water interchange between the Atlantic and the Mediterranean takes place in the
two channels, one to the north and one to the south of "The Ridge" and follows
exactly the same principles as those outlined above. Complete scientific use has been
made of the available observational data by Schott (1928 b). Longitudinal tempera-
ture and salinity sections are presented in Pt. I, p. 182, Fig. 83 for the transitional
period between spring and summer during which more or less mean current conditions
prevail. Seasonal variations in the velocity and extent of the upper current towards
the east and in the lower current towards the west are quite large. In the winter months
(including April) the thickness of the upper current is small, while that of the lower
current is rather large. During the summer months (to the end of October) the thick-
ness of the upper current increases by 80-100 m and that of the lower current is de-
creased correspondingly. During this part of the year the upper current must make up
the evaporation deficit in the Mediterranean. From the limiting position of the
boundary layer between the two water types it can be concluded that its annual varia-
tion west of the rise is of the order of 70-80 m, while east of the rise correspondingly
100 m or little more. The current boundaries also vary by similar amounts. The water
boundaries and the boundary between the currents do not coincide, but mixed
Mediterranean water is carried back with the upper current over almost the whole of
the area. Figure 245 gives a schematic representation of this. According to de Buen
(1926), Mediterranean water does not pass westward over the Gibraltar rise in the
deep layers, but is piled up on the eastern side and is carried backwards into the Medi-
terranean by the upper Atlantic current with an upward motion. Analysis of the ocean-
ographic series observations leaves no doubt of the incorrectness of this view of
de Buen.

530

Currents in a Strait

531

100

E 200

. 300

1400

500

Mediterranean seo

0
100

200

^- 300

Q 400
500

, . • Boundary between different

,^ — water masses

-^''"^_ r r /^

.■■' ^-^ . ~ Limit between currents

\ of different (jirection

Summer

Fig. 245. Schematic representation of the water type and of the current Hmit in the inner
region of the Strait of Gibraltar (according to Schott).

The few sporadic current measurements that have been made in the Strait of Gib-
rahar are in good agreement with the currents deduced from the longitudinal sections.
The upper current towards the east is particularly strong in the middle of the strait and
on its southern side. In the bays on both sides of the strait there are large vortical
movements ("neer" currents). The main current is considerably affected by wind and
tides and persistent easterly winds may even stop at time the inflow into the Medi-
terranean. The mean velocity of the upper current core according to Nares (1872) is
34 cm/sec ; ebb and flood superimpose the mean velocity and this results in a velocity
of +57 cm/sec giving an eastward flood current of 91 cm/sec and a westward ebb
current of 23 cm/sec. The currents are strongest along the southern edge of the deeper
southern channel and may reach as much as 210 cm/sec. The preference for the south-
em side, due to the Earth's rotation, can also be seen by means of the thermo-haline
cross-sections which show the Atlantic water of low salinity deflected to the right
along the African side.

Measurements made by the "Michael Sars" expedition (Murray and Hjort, 1912,
p. 290) give the vertical current profile shown in Fig. 246. The current boundary lies
at a depth of 142 m. Equation (XVI. 13) allows a computation of the average down
slope of the physical sea level from the Altantic Ocean to the Mediterranean and one
obtains as an average value 0-6 cm in 100 km, while the current boundary surface
rises by about 15 m in 100 km. In spite of the simplifying assumptions in the theory
there is satisfactory agreement between observations and theory. Direct current

532

Currents in a Strait

measurements have been made by Idrac (1928) on the vessel "Pourquoi-pas" in
March 1927 to the south of Tarifa. These gave the following values (depth 600 m).

Depth (m)

100

200

300

400

500

Current towards

NE 1/4 E

NE1/4E

W 1/4 N

W1/4N

Velocity

(cm sec-^)

The current boundary surface very probably lies at 1 50 m v/hich is about the same
depth as that found above. This can be compared with more recent investigations by
Menendez (1956).

1/ cm/^ec 1/, cm/sec

O 20 , 40 60 80 100 0 20^ 40 60

-•^,

200

250

300

350

Fig. 246. Vertical current distribution in the Straits of Gibraltar and Bab el Mandeb;
1 1 1 , according to the observations; • — • , according to the theory.

Conditions in the Strait of Bab el Mandeb are essentially similar. The temperature
and salinity distribution are given in Pt. I, p. 182, Fig. 84. Rather early current
measurements have been made by Gedge (1898) in the Perim Strait at the surface and
at 192 m, and they indicate a strong inflow with a velocity of at least 2-2-75 nautical
miles per hour at the surface. The current intensity decreased rapidly with depth and
showed a reversal in direction at 130-140 m; the speed of the lower current was
variable between 1 and 3 nautical miles per hour. The research vessel "Arimondi"
in 1924 made a 15-day measurement at almost the same place and a harmonic analysis
of this data was made by Vercelli (1925). The results for the basic current are given
in Table 143.

Table 143. Velocities of the basic current in the Strait of Bab el Mandeb

{March 1924) sea bottom at 175 m; depth of boundary surface at 100 m

(+, inflow from the Gulf of Aden; — , outflow from the Red Sea)

Depth (m)

100

130

150

Velocity (cm /sec)

+ 66

+ 59

+40

+ 1

-30

-68

Currents in a Strait

533

This also gives a good fit between observed values and the theoretical current
profile (Fig. 246); the low value at 130 m depth is apparently due to the uncertain
elimination of the tides. From the Gulf of Aden to the Red Sea the sea level falls
M cm in 100 km and the internal current boundary surface rises about 40 m. Since
the sea bottom from the sill out into the Gulf of Aden falls almost steadily from 150 to

0
20
40

60
80
100

"a.

'WM//;;,

1 1

I-.

cm/sec -4
To the South

4 8 12

To the North

Fig. 247. Vertical stratification of the basic current in the Strait of Messina (according to the
observations of the anchor station of the R.N. "Marsigli", 16-30 August 1922).

about 350 m, the internal boundary surface will also decline in the same direction, so
that the value given above is merely the deviation from the bottom slope. Because of
changes in the direction of the wind from the winter monsoon (east to south-east) to
the summer monsoon (north-west to west) the currents in the Strait of Bab el Mandeb
are subjected to oscillations with a semi-annual period. In the winter the inflow into
the Red Sea is a wind-drift current of strong permanence in speed and direction but
larger variations are to be expected during the summer monsoon.

{d) Strait of Messina

The smallest cross-section in this strait between the Ionian and the Tyrrhenian Sea
is at the northern end of the strait. Here it has a cross-sectional area of only \ km'*
with a mean depth of about 80 m and a maximum one of about 120 m. From this sill
the sea bottom slopes downward, uniformly and rather rapidly in valley form on either
side. At the northern outlet the mean depth is 140 m and towards the south it is already
about 900 m at about 30 km south of the sill. Since the water of the Ionian Sea is
heavier, the current flows from the Tyrrhenian Sea into the Ionian Sea in the upper
layer and in the opposite direction in the lower layer. Current measurements over a
15-day interval by the research vessel "Marsigli", at different depths down to 90 m
at a section in the narrowest part of the strait, have been analysed harmonically by
Vercelli (1926). Figure 247 shows the vertial current profile. Down to a depth of 30 m
the current flows to the south, below this, down to the sea bottom, to the north.
The velocities are small, in accordance with the low density differences, with on the
average about 4-3 in the upper current and about 9-3 cm/sec in the lower current.
Strong tidal currents are superimposed on the basic current and there are also strong

534 Currents in a Strait

disturbances due to atmospheric pressure and wind variations (velocities of up to
50 cm/sec). The current transport in the strait can be estimated from the Knudsen
relations (see Chap. XII. 5). For cross-sections at the narrowest point (Punta Pezzo-
Ganzirri) and at the rise of Punta Pellaro to the south of this, the mean salinities are :

s = 37-9, z = 38-4 and s' = 38-5, z' = 38-75%o.

The equations (XIII. 19) then give as an approximation / = u and /' = m' = 2/. Under
stationary conditions the transports will be the same in upper and lower currents, but
the transport through the southern cross-section is twice as large as that over the rise.
Thus only about half of the water of the lower current entering the southern part of
the channel flows over the sill to the north, the other half is carried back in the upper
current mixed with Tyrrhenian water. A corresponding calculation for a cross-section
to the north of the sill shows that part of the Tyrrhenian water entering the strait from
the north mixes with the lower current and is carried back into the Tyrrhenian Sea.
There must therefore be large turbulent mixing processes within the strait (see Vol. II).

4. External Influences (Bottom Topography, Tides) on the Oceanographic Conditions
in Sea Straits

The normal steady current conditions in sea straits may be modified by external
circumstances. It has already been mentioned that the atmospheric pressure and winds
have considerable influence. Some idea about these influences can be obtained by
simple numerical calculations. Besides these there are also other effects, especially
that of the bottom topography of the strait and also those of tides, which penetrate
from both sides from the open ocean into the sea strait and give rise to special current
phenomena there. These latter phenomena will be discussed later in Vol. II, but it
seems to be of advantage to mention these processes in connection with the funda-
mental phenomenon of water interchange between two seas already here.

(a) Disturbances Due to the Sea Bottom Configuration

The influence of a wave-form bottom topography on a horizontally flowing current
can be understood quite easily by means of theoretical computation, provided the
bottom relief can be expressed in the simple form

>'=-/? + y cos KX, (XVI. 14)

where k = 2irjL is determined by the known wavelength of the bottom waves. The
current with a velocity c over such a bottom will also take a wave-form, i.e. all layers
from the bottom to the surface will follow the bottom topography but with an ampli- •
tude decreasing with distance from the bottom. The sea surface itself will be a stream
line and its profile is determined from

^ ^ cosh Kh{\ - (glKC^) tanh k/j) ' (XVI. 15)

The denominator will be positive or negative according to whether

c ^ {(glK) tanh KhY'K

This expression is, however, the velocity of propagation of a wave in motion/ess water
of constant depth h. If the dimensions of the bottom waves are large compared with

Currents in a Strait 535

the depth, which is usually the case, then this critical velocity of propagation reduces
to the value Vgh. The stationary current waves in moving water will have exactly
the same form throughout the entire water layer as the bottom wave if c > ■\/sf^'->
if, however, c < \/gh then above a certain height it will be inverted, that is, above a
rise in the bottom there will be a depression of the water level and above a depression
in the bottom there will be a lift of the water level. If the velocity c is exactly the velocity
of free waves then resonance will occur and in this case the frictional forces will be
decisive. In all cases occurring in nature \/gh is always several times larger than c
and the stream lines show the wave-form of the bottom with decreasing amplitude
and only up to a certain height ; above this level of no horizontal motion the wave is
inverted, but the amphtude is so small that these waves will scarcely be noticeable. It
cannot be excluded that many of the vertical displacements in isotherms and iso-
halines, which are always found at the same place, may be due to effects of this type
produced by bottom disturbances.

In stratified water conditions are different, especially when there are well-
developed transition layers. Under certain conditions the disturbance by the bottom
relief may be shown in amplified form at a boundary layer; it may even be larger than
the disturbance causing it, while the surface of the water remains almost entirely
unaffected. Theoretical treatment is also possible in this case (Defant, 1923). If the
thickness of the upper layer is hi and that of the lower layer /zg, resonance (enlarged
amplitude of the stream-line waves) will occur at two values of the current velocity.
If the total depth of water h^ + Ag is small as compared with the wavelength of the
bottom disturbance these values are given by the equations

c,-Vk(h + h.)} and ., = y[(l-^j)j^J. (XVI.I6)

The first value Cj already for small depth is many times larger than any values found
in nature. Cg is the velocity of propagation of internal waves at the internal boundary
surface (see Vol. II) and may be so small that it can be quite close to the observed
current velocities. At these values the boundary surface will show the greatest varia-
tions while the sea surface remains almost undisturbed. For example, choosing pg — Pi
= 10~^, P2 = 1-028 then for larger h^ and hi = 50 m Cg will be 0-7 m/sec. Values of this
order are frequently found in sea straits and it can be expected that at corresponding
current velocities there will be large stationary vertical displacements in the density
transition layer.

The currents in the two water masses in sea straits usually have different velocity
values and are of opposite directions. This case can also be treated theoretically. If
the thickness of the upper and lower layer is small compared with the wavelength of
the bottom wave and their velocities are c„ and Ci, then the conditions for large
stationary boundary waves is given with sufficient accuracy by

c] hi -{-clhl=(^l- ^^ g hi h,. (XVL17)

A good example of this case is shown in the longitudinal density section through the
Bosphorus in Fig. 241. The isopycnals clearly follow the outline of the bottom.

536

Currents in a Strait

The disturbances are obviously due to this since the equation (XVI. 17) is approxi-
mately satisfied. Putting p^ =l-028, Pa — Pi as approximately 15 x 10"^ h-^ = 25 m
and h^ = 45 m, and since by observation c„:C}. = 2 then equation (XVI. 17) gives the
critical velocity of the upper current as c„ = 1-77 m/sec, while the observed values lie
between 1 and 2 m/sec.

The upward bulging of the boundary layer in the Strait of Gibraltar and the Strait
of Bab el Mandeb is undoubtedly due to the passage of the current over the rise in the
middle of the strait. Bulges such as these do not occur in a plane channel.

{b) Tidal Effects

Since tidal currents entering a sea strait affect the whole water mass from the sea
surface down to the bottom, the ebb and flood currents will be superimposed on both,
upper and lower currents, either reducing or accentuating them.

Since these currents flow in opposite directions the current profile will show rapid
changes over a tidal period. An example can be taken of a strait 300 m deep with
current reversal at 200 m in which the upper current flows east and the lower current
flows west; the upper current is assumed with a surface velocity of 100 cm/sec decreas-
ing parabolically with depth, while the lower current is supposed to increase below
the boundary surface. The amplitude of the tidal current may be 86 cm/sec and the
phase 3 moon hours (ebb towards the east at 3 h and flood towards the west at 9 h).
The current structure over a total tidal period is then shown schematically in Fig.
248. At 3 h there is a current directed to the east through the entire water mass with a
maximum at the surface; 6 h later conditions are almost reversed and the current is
directed towards west with a maximum at the bottom.

In addition to this direct influence there is also a second one affecting the boundary
surface. This will perform periodic internal vertical displacements initiated by the tidal

Fig. 248. Isopleths of the current velocity (cm/sec) in a water column during a total moon
period with a superposition of the basic and tidal current. (Type of currents in the Gibraltar

Strait.)

Currents in a Strait 537

rhythm which will also give rise to variations in the oceanographic factors. It can be
shown that the small periodic variations in the slope of the sea surface, produced by
the passage of the tidal wave, will be accompanied by waves at the internal boundary
layer of corresponding form, but of increased amphtude which will affect the normal
water interchange between the two seas.

A disturbance of the internal boundary surface in a sea strait due to a periodic displacement of the
sea surface (tide) can be treated theoretically in a simple way. The equations of motion for both layers
can be obtained from equation (XVI. 11), taking the local accelerations du^'dt and du^ldt, respectively,
into account. A periodic displacement of the sea surface can be given the form

Ci = ^acosA^exp(/par), (XV1.18)

where the variation in the surface gradient has a wavelength A, a period a and amplitude a. These
periodic vertical displacements of the sea surface give rise to corresponding variations in the upper
and lower currents of the form

Ml = v{z)a sin Ajc exp {/ {■r}ihy)at) and u^, = ^{z)y sin Xx exp {/ (r)lh^)at)} (XVI. 19)

and these will be associated with a period vertical displacement of the boundary surface

$2 = 1^7 cos Aa- exp ( / ^ at) (XVI.20)

hlg

v{z) and <P(z) fix the vertical velocity distributions in the upper and lower currents, respectively, and
follow from the differential equations of motion mentioned above and the corresponding boundary
conditions, y in equation (XVI.20) is the magnitude of the variations of the internal boundary surface ;
its value is given by

Pj—a\\-^-^M\. (XVI.21)

— Pi L Pi J

Pi— P\ L Pi

Since M (see p. 520) is always negative, it is clear that the variations of the boundary surface will
always be the reverse of those at the sea surface, and since y is inversely proportional to the difference
in density between the two water types they will be many times (of the order of about 1000) greater
than the latter.

Variations of this type appear in all extensive series of observations. Schott (1928)
has investigated the observations made by the "Dana" expedition in the eastern part
of the Strait of Gibraltar and obtained the results shown in Fig. 249. Values for the
layer from 100 to 200 m were combined to eliminate the irregularities in individual
values and to accentuate the connection with the tidal period. The isotherms and
isohahnes rise and fall in time with the sea surface tide at Gibraltar; here the oscilla-
tions of the internal boundary reach the large value of 70-80 m. Similar results were
obtained at the "Dana" station for 14-15 July 1928 by Jacobsen and Thomsen
(1934) where the 37%o isohaline had an average amplitude of 66 m, at neap tides
42 m, and at spring tides 90 m.

Similar vertical oscillations in the density transition layer were found at the 15- day
anchor station in the Strait of Bab el Mandeb ; they follow the rhythm of the tidal
currents and have amplitudes of up to 100 m. In this case there is a phase shift of 3 h
between the current curve and the thermo-haline curve. This is shown in a particularly
clear manner by taking the mean of 5 semi-diurnal periods. (Table 144.) The extreme
values of temperature and salinity occur at the times of current reversal. Here, as in
the Strait of Gibraltar, the main cause of the variations in the density transition layer
is the passage of tidal waves. These quite large displacements of the boundary layer
can also be explained quantitatively by the theory. Assuming the amplitude of the

538

Currents in a Strait

1 < 1 1 1 1 ! 1
High water and low waterot Gibraltar

-K—

_(5°20'W)

•\

■—

^4-^'^'

♦^^^

T—

368
369
370
37 1
372
37 3

1
1

/>!

\\^

r-.

[

37 t^

376
377
37 8

/ /

f /

\'\ i

■^ITi

■ ^,

18 20 22 0 2
8X1921 9X

8 10 12 14 16 18 20 22 0 2

lOX

Time, hr

152

15-0

146

144

142 o

140

138

136

134

132

13 0
128

4 6

Fig. 249. Strait of Gibraltar: periodic oscillations in the mean salinity and mean tempera-
ture of the layer 100-200 m according to the observations of the "Dana" St. 1138 (5° 30' W.)

(according to Schott).

Table 144. Tidal current and periodic variations in temperature and salinity

in the Strait of Bab el Mandeb aMOO m depth

(Five semi-diurnal moon peroids)

Moon hours

0 1 2

3 4 5 6 7 8

9 10 11

10^ nautical
miles per hour

+ 89 +87 +67

-7 -61 -112 -102 -71 -36

+ 36 +100 +105

Flood current

Ebb current

Flood current

Salinity 36 "/oo

0-84 0-91 0-96

0-991 o-99t 0 89 0-78 0-69 0-59

0-52* 0-60 0-66

Temperature
25 C

0-28 0-21* 0-24

0-25 0-23 0-25 0-51 0-69t 0 69t

0-55 046 0-39

* minimum; t maximum.

oscillation in the sea surface slope as 10 cm in a model of the Strait of Gibraltar between
Tarifa and Gibraltar, gave an internal boundary oscillation with tidal period and
amplitude of 110 m which is in agreement with the order of magnitude of the observed
values. Strong well-developed internal tide waves were also found at the 15-day
anchor station in the Strait of Messina. This case is of particular interest because the
wave here reaches the limits of stability characteristic for such waves and at times even
exceeds it (see Vol. II).

5. Processes in Estuaries (River Mouths)

River water flowing into the sea gives rise to compensation currents along the river
bed, which show similarities to current processes in sea straits. Ekman (1876) in an

Currents in a Strait

539

investigation of Swedish rivers found that the outflow of river water in the estuary
was accompanied by an inflow of sea water in the lower layers. Thus, at the mouth of
the Gotaelf into the Elfsborgsfjord, there was a strong compensation requirement for
the outflowing surface water which could not be satisfied by inflow from the sides. It
therefore gave rise to upweUing motions from below. The consequent reverse deep
current was clearly shown by the salinity distribution at different depths and could also
be shown experimentally by drift buoys. The rising water was both more saline and
more transparent than the sewage-laden river water. Figure 250 shows the salinity distri-
bution along a longitudinal section; the upstream directed lower current is demon-
strated clearly by the 20%o isohahne.

Fig. 250. Vertical distribution of salinity in the river mouth of the Gotaelf. (I) 5 August

1875; (II) 19 February 1890.

A theoretical investigation of the occurrence of lower currents of this type in river
mouths (estuaries) was made by Ekman (1899) using principles similar to those used
in the theory of currents in sea straits. He found that under normal conditions there
were no currents carrying sea water upstream, but that such a current was formed
immediately if there was a tangential force acting on the sea surface. The shallower the
water, the greater must be the tangential pressure in comparison with the surface
(river) velocity in order to allow for the generation of a compensation current in the
deep water. River water entering an estuary flows on top of the sea water partly
because of its inertial momentum and partly because of its lower density. It thus
exerts the tangential pressure on the lower layer which favours the compensation
current.

The momentum and the density are apparently, however, of less importance than
the density difference between the upper and lower layers and turbulent mixing of the
two water types.

540

Currents in a Strait

This compensation-current phenomenon probably occurs at the mouths of most
rivers, especially those carrying large quantities of water but no accurate systematic
investigation has been made of these processes.

The situation is different for processes in the sea remote from the mouth of a river.
These are easily handled theoretically (Takano, 1954, 1955) and the stratification in
the sea, the vertical and lateral mixing and the turbulence of the current can be taken
into account.

Taking a vertical coast as the j'-axis and at this coast a river mouth where
— /<>'</ from which the river water with a constant velocity Uq flows into the
open sea at right angles to the coast, then, neglecting inertial terms and any tidal
effects present, the equations of motion and the continuity equation will be

- pfv =

dp __ 8 / 8u ,

8y+^^'^'+ 8:

8z\ ' 8z
8

f(^'S)-

8pU ^PV ^r.

8x ^ 8y

(XVI.22)

Ah and A^ are the lateral and vertical eddy viscosities and /is the Coriolis parameter
which can be assumed constant.

Assuming that the stress both at the surface (z = — i) and at the bottom (z = d)
vanishes and introducing the volume transport {p '^ \) one obtains

pu dz and M

-z

y= \ pv dz

(XVI.23)

and putting P = \ P d^ gives from equation (XVI.22)

AnVm^^fMy =

8^'

AnV^My~fM, = ^^,

8M, ^My^

8x "^ 8y

If the stream function is taken as usual

84,

then from (XVI. 24)
whereby

M^.= -j-y ^"d ^^ ^ + e:^ '

vv -= 0,

g4 g4 g4

V * = !- 2 I —

8x^^ dx^8y^ ^ 8y^

(XVI.24)

(XVI.25)
(XVI.26)

Currents in a Strait
is the biharmonic operator. With the boundary conditions

at .V = 0 and — I < y < I: M^--- Mq

at .V = 0 and

l> y> I: M^ = 0,

541

(XVI.27)

where Mq is the volume transport of the river flow at the mouth (which is assumed to be
uniform), the solution of (XVI.26) will be given by

M„

0 = i^»<|(^ + /)tan-i-
Equation (XVI.24) thus gives

+ / V - /
(v - /) tan-1

X X

(XV.28)

/^-^!-/|(>- + /)tan-4-'-(.-/)tan-^-^'

+ 2An

y + i

y-l

-v' - Cv + 0^ '^'' + (y - 0'

(XVI.29)

H 1 1 1 1 1 1 1 1 h

FiG. 251. Spreading of light river water off the mouth in the ocean for different values of
the horizontal exchange, (a) R = 1/500; (b) R = 2/500; (c) R = 4/500; (d) R = 8/500;
(e) R = 16/500; (/) R = 32/500. Dashed curves: /= 0 (zero Coriolis parameter, non-
rotating system) (according to Takano, 1955).

542

Currents in a Strait

The vertical density distribution is assumed to correspond to that of the Reid model
(1948)

P = Po; - ^ ^ z ^ h; p= pa-Ap e^-'^^ (h ^ z ^ d) (XVI.30)
where

Ap= pd- Po and p = pd {d S z).

This corresponds to a homogeneous top layer of thickness h with a lower layer in
which the density increases to p^. Then as a first approximation

81: 2Ap 8h dP 5gAp 8h^

— ■ /->-' — and — '-^ ■ —

8x Pq dx dx 2 dx

Analogous equations will apply for y and furthermore

(XVI.31)

/l2=-

5gAp

(XIV.32)

The integrated pressure P can be taken to represent the thickness of the upper homo-
genous layer. Putting /= 0 in equation (XVI.29), that is, neglecting the CorioHs force
gives

Fig. 25la. Schematic representation of the spreading of river water in the ocean off the

river mouth.

Currents in a Strait

543

^/ = o =

lAnM^ f y + l

y-l

^ \x'' + (j + 0' x^ + iy- 0'

AAj^MJ i X

v2 _ j2 _|_ /2

[.^2 + 0 + /)2] [.x2 + (j - O^]/- (XVI.33)

If >'2 — x^ = /2 then h vanishes, that is, the lighter river water fills only the volume
between the hyperbolic branches y^ — x^ = P and jc = 0. The river water flows as an
upper layer over the lower layer, spreading out laterally between these hyperbolic
branches. The first term in (XVI.29) modifies this simple symmetrical spreading of the
river water on top of the lower water. This is purely an effect of the lateral and vertical
mixing process ; it causes the homogeneous layer to be deeper on the right-hand side
and shallower on the left-hand side. The inflow is thus directed to the right in the
Northern Hemisphere. Figure 25 1 shows the limits of the river water for the different
cases

A^ 500

2
500

4
500

500

16
500

and

32
500

where the dashed curve is for / = 0 (non-rotating system).

Table 145

2/ in m :

200

5.0

106

2.0

106

1.25 X

106

6.2

105

3.1

105

1.6

105

600

4.5

10^

2.2

107

1.1 X

107

5.7

106

2.8

106

1.4

106

1000

1.26

108

6.2

107

3.1 X

107

1.6

107

7.8

106

3.9

106

2000

5.0

108

2.0

108

1.25 X

108

6.2

107

3.1

107

5.6

10'

Exchange coefficients for the cases shown in Fig. 257 are contained in the following
Table 145 for a corresponding river mouth width 2/ and for/= 10~^ sec~^. The Coriolis
force deflects the seaward flow towards the right and gives rise at the mouth of a river
in the Northern Hemisphere to a water level sloping from the right bank down to the
left bank. For the lateral exchange coeflicients found in practice, 10^ to 10^, and for a
river mouth width between about 300 m and 1 km there will be quite a sharp deflec-
tion to the right (approximately as in curves d to/). The flow of river water into the
sea at the mouth of a river is shown schematically in Fig. 251a and conditions actually
found in nature will probably correspond reasonably well to this.

Chapter XVII

Effect of Wind on the Mass Field and
on the Density Current

Under stationary conditions all the forces acting must be in equilibrium and the mass
distribution must be adapted to this equilibrium if it is to be maintained. In this case
it is not possible to distinguish between cause and effect; there is usually a mutual
adjustment between the internal field of force and the current present. If there is a change
in the field of force then there must also be a subsequent change in the current;
conversely if there is a change in the current there must be a rearrangement of the
field of force until equilibrium is again restored. These circumstances should be kept
in mind for an understanding of the way in which wind influences density currents.

1. A Limited and Stratified sea

Conditions in a limited trough-like sea shall be considered first. Work in this
direction has been done by Palmen (1926, 1930 a, b and with Laurila as co-worker,
1938) for the Gulf of Finland and the Gulf of Bothnia, principally in particular
cases which are only able to give some insight into the mechanism of the processes
which occur. The influence on the water stratification occurs as follows :

We assume at first no wind at all over a barotropic sea ; the isosteric surfaces and
especially the transition layer between the top layer and the deep water will then
follow level surfaces (Niveauflachen). If a wind starts, the surface waters are forced
to move first in the direction of the wind, but the Coriolis force will soon produce a
deflection to the right (Northern Hemisphere) and a piling-up of the water along the
sea coasts. In an elongated ocean bay the final result will be a current predominantly
occurring along its longer axis. In addition to the wind-generated current in the top
layer a gradient (Stau) current is then added in the deeper layers due to the piling up of
water which will flow approximately in the opposite direction. Thus a vertical circula-
tion in a longitudinal direction is set up and an equilibrium state is present in which the
transport due to the surface current is exactly balanced by that of the deep current.
This quasi-stationary state of the current is fixed at each level by an equilibrium
between the gradient force, the Coriolis force and the frictional force. Since a stronger
current is only possible along the longitudinal axis of the bay it follows that the direc-
tion of the gradient force usually does not coincide with the direction of the current
itself but the deviation will not be great. In addition to the principal gradient in a
longitudinal direction in the layers above and below the level of current reversal (layer
of no motion) there will also occur smaller components of the pressure force acting
at right angles to the direction of the current. These will be largest at the surface and

544

Effect of Wind on the Mass Field and on the Density Current

545

will decrease with depth-changing sign at the layer of no motion. This will modify the
mass field which then can no longer remain barotropic. The isosteric surfaces must slope
transversally ; the mass field becomes baroclinic. The structure of the associated density
current can be computed by means of ordinary methods from this mass field. The
primary factor will now no longer be the water stratification but rather the current,
while the water stratification can be regarded as a consequence of this current.

Palmen investigated data for the Gulf of Finland for steady westerly and steady
easterly winds and distinguished between a west type and an east type. He deduced
mean mass fields over a cross-section for these two cases from the large amount of
data available. In the east type the lighter surface water lies in a wedge-form at the
Finnish coast with the isosteres sloping downwards from south to north, while in
case of the west type conditions are reversed. Figure 252 shows the distribution of density
for the two opposite types. The interpretation is simple: the west wind produces a
drift current in which the transport is directed towards the Estonian coast where the
lighter surface water will pile up. For an east wind the opposite occurs. Palmen has
demonstrated the reality of these changes in sea level between the northern and southern
sides out of observations of water level in Hango, Reval and Helsinki. For the east

Estonio

Finland

J 100

Fig. 252. Normal density distribution in the cross-section Aransgrund-Kokskar (Fennic
Bay, 25"' E.); at, values. , east type; , west type (according to Palmen).

546

Ejfect of Wind on the Mass Field and on the Density Current

type the difference in water level was 4-4 cm and for the west type this difference is
3-4 cm. The absolute velocity of the wind-generated surface current will thus be for
the east type 6-9 cm/sec towards the west and for the west type 5-3 cm/sec towards
the east. Current measurements give 7-5 and 6-0 cm/sec, which is in good agreement.
The relative changes in velocity with depth can be calculated by ordinary methods
(equation XV.20) from the mass field and can then be converted to absolute velocities
using the surface velocities given above. Table 148 containing these values shows clearly
the division of the current structure into two layers; at the middle of the Gulf of
Finland the current reversal is at a depth of approximately 27 m. It changes in a
corresponding way towards the Finnish and Estonian coasts. The calculated values
are a little too large, since friction has been neglected, but otherwise are in satisfactory
agreement with observed values. In some special cases for a strong wind and steeper
inclination of the isosteres in the transverse section, the velocities are much greater
(for instance, 7 October 1936; surface velocity 23-5 cm/ sec) and the layer of no motion
occurs at greater depth (about 35 m) in full agreement with the observed values.

Table 146. Current stratification for different wind directions in the Gulf of Finland
(according to Palmen) (positive sign towards west; negative sign towards east)

Depth (m)

Velocity (cm/sec)
For east type
For west type .

+ 7-3
-5-3

+ 51

-3-7

+ 1-8
-11

-0-9

+ 1-3

-3-3

+ 3-7

-4-3
+4-6

-5-3
+ 50

-5-3

+ 5-3

When the wind is in a direction other than directly east or west only the eastern or
western component will have any effect. The inclination of the isosteres in the trans-
verse section will therefore be correspondingly less and the number of solenoids will
thus be reduced and must therefore show a dependence on the wind direction.

The rearrangement of stratification caused by the wind in an elongated oceanic
region will thus proceed in the following way:

(1) A steady wind with a component along the longitudinal axis of the sea will
originate a vertical circulation; this will be made up of a drift current in the top
layer and a corresponding gradient current in the deep water.

(2) This current system will produce a vertical transverse circulation which in turn
will give rise to an inclination of the density transition layer and of the isosteric
surfaces, that is, the longitudinal circulation produced by wind will give rise to a
solenoid field at right angles to this circulation. The strength of this field will be a
function of the wind influence. When an equilibrium state is reached this cross
circulation will vanish.

(3) A transverse slope in the physical sea level will develop at the same time and its
intensity will also be dependent on the wind.

(4) From the solenoid field and the transverse slope of the sea surface the current
structure in a transverse section can be calculated. In a steady equilibrium state the
slope of the internal boundary surface in a two-layered sea will be greater than that

Effect of Wind on the Mass Field and on the Density Current 547

of the physical sea level in the ratio Pi:(p2 — Pi). It is easily shown that this slope is
given by

•_ ^

g{p2. — Pi)hi

where pi and p, are the densities of the top and lower layers, respectively, h^ is the thick-
ness of the top layer when the system is at rest and T is the shearing stress of the wind.
The deep water is assumed to be motionless. This relationship has the same form as
the equation (XIII.45) which gives the piling up of water by the wind (Windstau) in a
homogeneous sea except that p is replaced by the density difference (pa — pi).
Hellstrom (1941) showed that in a stratified sea with two layers the piling up of
water by the wind differs markedly from that in homogeneous water and that the effect
of the wind is larger. The wind stress calculated from equation (XIII.45) (p. 419) is
much too large, and the less the depth of the discontinuity layer the greater is the error.
Palmen's investigations, however, showed that the changes in water level in the Baltic
due to the effect of the wind are almost independent of the water stratification. This
contradiction was resolved by Palmen (1941) by estimation of the time required to
establish an equilibrium state. This time required is very large, of the order of several
days, while only a few hours are needed to produce a piling up of the water similar
to that for homogeneous water. Usually, the wind direction does not remain invariable
for a longer time to allow the slopes of the discontinuity layer and the sea surface to
reach a steady state. Initially, the piling up of water by the wind in a stratified sea is
approximately the same as in a homogeneous sea. However, the longer the duration of
the wind the closer is the approach to the Hellstrom values. The equation (XIII.45) can
thus be used in almost all cases for the calculation of the wind pressure, although
strictly it is valid only for homogeneous water.

Fjelstad (1946) has made a thorough theoretical examination of steady currents
in a stratified water contained in a wide channel and has obtained results in complete
agreement with the observations.

The transverse circulation is usually connected with another important pheno-
menon. In a sea of sufficient width a strong wind may produce an inclination of the
density transition layer sufficient to bring the deep water to the sea surface. A rapid
fall in temperature will then occur and an increase in salinity in a long band along the
coast to the left of the current (Northern Hemisphere). The phenomenon of "cold
upwelling water" along an extended coastline has previously been regarded largely as a
direct result of an offshore wind (land wind) (Sandstrqm, 1922; Krummel, 1911,
p. 536 and following), forcing the deep water upwards to the surface at the lee coast
while the surface water is forced downwards to deeper layers at the windward coast (luv-
coast). Besides this direct effect, the effect of earth rotation in the above senses, seem
however, of more importance. In the Gulf of Finland and in the Baltic (Mae, 1928) the
upwelling of cold water found during strong persistent longitudinal winds gives
support to the importance of the indirect wind effect,

2. General Conditions in the Open Ocean

These are essentially the same as in channel-form elongated oceanic regions. The
efiFect of the wind is mostly restricted to a more or less broad band of the sea surface,
and outside this area the water is either motionless or subject to the effect of a wind

548 Ejfect of Wind on the Mass Field and on the Density Current

from another direction. Thus, for example, in a broad band of an oceanic region with
vertical increase of density and forming a channel around the earth in the Northern
Hemisphere, conditions will be more or less as follows.

If there is a persistent wind in the direction of the channel the immediate effect of
the drift current (westerly wind) is to transport lighter surface water to the right
(south) side of the channel. In the top layers the isosteric surfaces can no longer be
horizontal and will adjust with an inclination from north to south in order to corres-
pond with the accumulation of lighter water on the right-hand side of the wind. A
solenoid field of this type will, however, produce a density current in the direction of
the wind in which the velocity will decrease with depth corresponding to a similar
decrease in the slope of the isosteres. At the same time, water will be piled up on the
right-hand (south) side of the channel and this will give rise to a gradient (Stau)
current in the direction of the wind. Its velocity will remain constant down to the lower
frictional depth. In this way the stratification will lead to a considerable complication
of the conditions and even more so if changes due to other factors (heating, cooling,
evaporation and others) must, too, be taken into consideration.

It is doubtful whether a gradient (Stau) current will be generated in such a current
system. The displacement of the water masses in the top layer, where the solenoids
are numerous and which is superimposed on deep water where the solenoids are few,
may proceed so that the isobaric surfaces in the deep water remain horizontal (see
discussion on p. 483 and following pages). If the effect of the water accumulation (rise
in physical sea level) occurring on the right-hand side of the wind direction (Northern
Hemisphere) on the pressure field of the deeper water is compensated exactly by the
baroclinic mass distribution of the top layer there will be no gradient (Stau) current.
In actual practice, the relationship between the topography of the physical sea level
and the mass structure of the upper layers is usually satisfied so that any deep reaching
slope current is improbable.

A complete theoretical treatment of the problem of currents in a baroclinic ocean
offers considerable mathematical difficulties, since it must take into account vertical
frictional effects, lateral mixing processes and boundary-surface conditions. In con-
nection with an investigation on the circulation of the antarctic circumpolar waters,
SvERDRUP (1933) has discussed the possibility of formation o{ o. steady drift current in
the presence of a baroclinic stratification of the water masses. He showed, in agree-
ment with the results of Ekman, that steady vertical circulations can hardly develop
in the ocean if only the effect of wind is taken into account. Due to the non-uniformity
of the wind field (divergences and convergences), and due to the boundaries between
different water bodies and the coasts, vertical circulations will be formed and will
produce changes in the mass field. However, since the density distribution in the sea
is usually a stationary one and apparently steady circulations still occur, it follows that
the effect of the vertical circulations produced by wind must be compensated by other
factors which affect the density. This gives emphasis to the great importance of these
factors for the development and maintenance of the oceanic circulation. Heating,
cooling, evaporation, precipitation and other factors thus take part indirectly in the
formation of the oceanic circulation. The convective sinking of cold waters in higher
latitudes plays an especially important part for the maintenance of vertical oceanic
circulations.

Ejfect of Wind on the Mass Field and on the Density Current 549

Ekman (1931) has drawn attention to a special effect of the wind on a given solenoid
field. In a top layer (the place where density currents occur) the isosteric surfaces are
assumed to rise from south to north (Northern Hemisphere; approximately the
conditions found in the Atlantic between 40° to 50° N. and 30° to 40° W.). In the
absence of wind there will be a density current directed towards the east. If now a
steady persistent wind gives rise to a drift current, thus altering the mass field, then, for
a northerly wind the total transport of the drift current will be directed to the west and
for a southerly wind to the east. The basic current therefore will be either retarded
or accelerated. An east wind blowing against the current will produce a transport of
the upper water to the north and will thus tend to even out meridional density differ-
ences, and in this way to decrease the velocity of the density current. If the wind blows
towards the west (as in the Atlantic over the Gulf Stream), then the upper layer will
be driven towards the south and the slope of the isosteric surfaces will increase. As
long as only the total system of surfaces without internal change is displaced towards
the south the strength of the density current, which is largely fixed by the horizontal
distances between the isosteres, will remain unchanged; however, under certain con-
ditions changes in inclination of these surfaces will also occur and the density current
will increase its strength. This is especially the case when the upper lighter water is
displaced by the wind, while the lower one remains unaffected. The wind blowing in
the direction of the density current, in addition to the generation of a drift current,
also has the effect of localizing the density current and may transform an otherwise
broad and slow current into a narrow rapid one, still with the same transport. Ekman
saw in this process an explanation for the narrowness to which the Gulf Stream is
confined in this part of the Atlantic. This peculiar phenomenon of a "river in the sea"
is in any case an argument in favour of such wind effects.

Another example of wind effect on the mass field is the boundary surface found
throughout the interior of the entire Antarctic Ocean which appears at the sea surface
of the ocean as the Antarctic Convergence Line (Southern Hemisphere Polar Front).
This boundary surface separates the heavier, colder, Antarctic water to the south
from the lighter but more saline water of the oceanic troposphere to the north. The
boundary surface has a slope corresponding to the density and current conditions. It
behaves like a solid wall (continental slope) and makes an Antarctic vertical circulation
possible. Figure 253 (Sverdrup, 1933a) shows a meridional density section at 30° W.
derived from the observations of the "Discovery" expedition. The boundary surface
meets the sea surface at 50° S. in the Antarctic convergence line. The topography of
the physical sea level and the 1000 decibars surface (both relative to the 3000 decibars
surface) are shown in the diagram above. These isobaric surfaces slope downwards
from north to south corresponding to the current flowing eastward in both water
bodies; this current must be stronger on the northern side than in the Antarctic water
to the south.

The cause of the formation of a discontinuity surface is not immediately apparent,
since the current flows exactly towards east in all latitudes and meridional current
components are required in order to produce and to maintain it.

Two factors favour the occurrence of a northward component in the Antarctic
water.

(1) The prevailing westerly winds, and

550

Effect of Wind on the Mass Field and on the Density Current

Fig. 253. Vertical section of density (a,) in the Atlantic Ocean along 30" W. between 24° and
58° S. Above: topography of the physical sea level and of the 1000-decibar surface (relative
to the 3000-decibar surface assumed as plane). A.C., Antarctic convergence (oceanic polar

front).

(2) the continuous supply of water with low salinity which is produced by melting
of the northward drifting pack-ice.

This second factor requires the presence of a thermo-haline circulation directed at
the surface from an area with high specific volume to another one with a low specific
volume. A circulation of this type is certainly present but the wind conditions are
probably the main cause (Deacon, 1934; Sverdrup, 1934Z)). In latitudes between
40° to 65° S. the prevailing wind is always westerly and gives rise to a drift current and
a consequent surface water transport to the north. According to meteorological obser-
vations the strongest surface wind in higher latitudes occurs between 50° and 60° S.
The water transport to the north is thus greatest between 60° and 50° S. and north of
50° S. is comparatively smaller. This gives rise to the formation of a convergence
line and a discontinuity layer in the mass field. The wind and its differentiation in a
meridional direction may also be considered the main reason for the intensification
and concentration within a narrow strip of the density current which would otherwise
spread out over a wider area.

3. General Relationships Between Wind and Currents

The investigation of steady currents produced by wind in a baroclinic top layer is
easily handled, since the deep water can be regarded as essentially motionless and the
wind field as quasi-permanent showing no changes with time or position. This allows
the eff'ects of both the vertical and horizontal eddy viscosities to be taken into account.
The equations of motion (XIII. 52) must then include terms for the horizontal eddy
viscosity, denoted briefly by h^ and h^. Integration of these equations over the entire
depth d and introduction of

//.

/^. dz, H, =

h„ dz and P

pdz

(XVII.2)

Ejfect of Wind on the Mass Field and on the Density Current 551

gives

^+/M, + r, + //, = 0,

~^-fM, + Ty + Hy = o.

(XVII.3)

Therein M is the vector of the mass transport (equation XII. 8, p. 376). To these
must be added the continuity equation for an incompressible fluid.

For a given value of T and ignoring the effects of the horizontal components of the
eddy viscosity the three equations (XVII.3 and 4) can be regarded as equations with
three unknowns P, M^ and My. Thus, in such a baroclinic current the total pressure P
and the mass transport M can be represented as functions of the wind stress.

Ehmination of P by cross-differentiation, taking into account equation (XVII.4)
and putting ^ = df jdy gives

(f-i)+^-.+rf-t)--

According to this vorticity equation the wind-stress vorticity must be balanced at every
locality by the vorticity of lateral mixing and by the term /SMy, which is the effect of
the change of the Coriolis parameter with latitude. This equation is reminiscent of the
equation (XIII. 59a) derived by Ekman who designated the term ^My the planetary
vorticity.

SvERDRUP (1947) and Reid (1948) have applied this equation to the equatorial
currents of the eastern Pacific Ocean which correspond closely to the above conditions.

The X-axis is taken pointing eastward and the >'-axis pointing northward. For the
trade wind belt it is possible to put dTy/8x = 0 so that neglecting lateral mixing,
(XVII(.5) gives

^My == - ^' (XVII.6)

and with (XVII.4)

and

M. = .-^(?^' tan 0 + i? ^^) (XVIL7)

2ajcos0\ej ^ 8y^ /

cP — dT^

ox dy

and

dP ^ 8^T^

^=-^^^^^^^'^ + ^-

Thus for X == 0, (at the north-south vertical boundary), M^ = 0 (integration limits
0 to Ax). The bars denote average values of the stress derivatives. The mass transports
Mg and My can be found directly from (XVII.3) if dP/dx and 8PJdy are known.

552

Effect of Wind on the Mass Field and on the Density Current

These equations have been tested by the "Carnegie" and "Bushnell" observations
of corresponding areas (approximately between 160° to 80° W. and 10° S. to 20° N.)
and showed good agreement with the values derived from the observations. The
theoretical values were calculated from the distribution of wind stress obtained from
the wind field given in oceanic climatological charts; thereby use has been made of
formulae (XIII.48 and 49). Figure 254 shows the excellent agreement between the ob-
served and theoretical meridional distributions oi APjAx and M^. It should be kept in

25M

V^'

fy\

i ^'^

-^._^^

■ /;(r1=-ff ton ^ jp-

+ r.

4^0ctNov grodients

JXCarnege stations

) ^

J/'t>Jov-Mar Qfodienls

- /

jr Carnegie a SusHnell

stations

° 0

/^5

° (

1 1 1

||'a/f(r)(dyn.cm-^

1 1

-20 -1-5 • -1-0

0 /

05 10

0 y

-5

-10

Fig. 254. Picture to the left: theoretical and observed values APjAx in two sections of
"Carnegie" and "Bushell" stations. Picture to the right: Latitude dependence of the longi-
tudinal mass transport computed by two independent methods. (M^ = eastward mass
transport in tons per sec through a column of 1000 m depth and 1 m width).

mind that the theoretical values are derived from mean wind conditions while the ob-
served values are based on some oceanographic stations made at different times of the
year. From these results it can be concluded that mass structure and mass transport
of the currents in the eastern equatorial areas of the Pacific can be regarded as a con
sequence of the average shearing stress of the air currents on the surface of the sea.
This conclusion should also be valid for the equatorial currents in other oceans.

4. Velocity Computations of Oceanic Surface Currents in the Equatorial Regions
from Wind Data

The currents in the equatorial regions can, as a first approximation, also be regarded
as the result of a drift current and a gradient current of the type described by Ekman.
However, at the equator itself the two components are indeterminate and the geo-
strophic approximation gives infinitely large values. In dynamic calculation these areas
must therefore be excluded. The question of how to calculate the currents in the
immediate vicinity of the equator from oceanographic data has been dealt with by
Weenink and Groen (1952), which gave an exact solution to the problem and by

Effect of Wind on the Mass Field and on the Density Current 553

TsucHiVA (1955fl, b) who made a second approximation to the geostrophic current
equation for/ = 0. For the surface velocity of a drift current and a frictionless gradient
current the equations (XIII.26 and 31) give

Tcos(ifj — 77/4)

- - ("A

rsin(iA-7T/4)
V(fpoV) ~^ fp<

(XVII.6)

where 0 is the angle between the wind stress and the direction of the ^-current; the
subscript zero refers to the sea surface. Indeterminate solutions are obtained from
(XVII. 6) for the equator. If an exact solution is required the eddy viscosity cannot be
taken as insignificant by comparison with the pressure gradient and the Coriolis
force. Only in this way there is an equilibrium between the wind stress, the pressure
force and the vertical friction in the equatorial belt. The simple equation of motion
(corresponding to (XIII. 23fl) and (XIII. 30) is now

where

V = Vjc ^ iVy and p ~ Po-

The boundary conditions are

L^] =.-T=-iT, + iTy) and y(z = O)) = 0 (XVII.8)

(XVII.7) is identical with

ry9

" av^b, (XVII.9)

where

a = — and d = [^ + ' ^

7] 7] \dx cy

If b{z) is known from observations then, taking equation (XVII.8) into account and
since a is independent of z this can be solved. To determine b{z) Weenink and Groen
used the Reid model (1948) which gives a good approximation for the equatorial
regions. This postulates a homogeneous layer of thickness h below which the density
of the water increases with depth according to an exponential function (see XVI. 30).
For this model (as in XVI. 31) one obtains

ldp\ Ap8h /8p\ Apdh

and the solution of (XVII.9) at the surface (z = 0) will be

„„ = Jl _ *« (, - 1+^%-H A (XVII.IO,

7]^/a a \ 1 + h\/a J

When the value of h\/a or of /is large the expression in brackets will equal 1 and
(XVII.IO) will be nearly equal to (XVII.6). It is thus apparent that at a latitude of 2° to

554 Ejfect of Wind on the Mass Field and on the Density Current

3° the value of h\/a is already large enough to allow equation (XVII.6) to be used
instead of (XVII. 10).

For a sufficiently narrow belt on both sides of the equator expansion into a power
series with respect to h\/a gives, neglecting higher order terms

t^o = :^ (^ - lb A + Ab,h^ + . . . (XVII. 1 1)

If lateral mixing is neglected (// = 0) the equations (XVII. 3) become

T = AP-^ifM (XVII. 12)

and (XVII. 1 1) with (XVI.31) becomes

Vo - 4boh^ + — ^ + . . . (XVII. 13)

Since M remains finite at the equator this gives finally by means of (XVI.31) and
(XVII. 12)

%Th
Vo=--F~ • (XVII. 14)

The behaviour of v^ can be illustrated in the following way. If the first term on the
right-hand side of (XVII. 10) is the drift current and the remainder of ^o is taken as the
slope current, then both components tend to infinity on approaching the equator,
but due to the coupling between these two components they behave in such a way
that their sum remains finite and approaches the vector (XVII. 14) as a limit of zero
latitude. The surface current, the wind stress and the surface pressure gradient all
have the same direction at the equator. Figure 255 illustrates their behaviour near
the equator.

wind

Fig. 255. The two components (vwind and Wgrad) of the current velocity (ftot) somewhere

near the equator. Exactly at the equator the vectors of the current velocity, the pressure

gradient Ap and the wind stress T fall all in the same direction.

More recently Yoshida (1955) has shown that the model used by Weenink and Groen
apparently leads to a solution involving a discontinuity in the vicinity of the equator.
This singularity originates in the assumptions of the model. A modification of the
model which seems more realistic in the light of recent observations appears to give
a reasonable solution.

Ejfect of Wind on the Mass Field and on the Density Current 555

The method of Tsuchiva is simpler. The equations of motion of the geostrophic
current are

where D is the geodynamic depth. All the quantities in these equations can now be
expanded into the Taylor series with respect to y and equation of terms of the same
power of V gives, putting /S = df jdy,

(-)^^0;,.„^-(P)^.„. (1)=0; .^0. (XVn..)

The distribution of D is easily found from oceanographic data. The east-west com-
ponent ?/o of the current velocity at the equator can therefore be obtained from the
second equation (XVII. 16) and the north-south component Iq is zero. At the same
time {8D/cx)o and (cD/8}^o must be zero. The oceanographic data show that these
conditions are fairly well satisfied in most cases. Values ot u and v near the equator
can be obtained by substitution of higher-order derivatives of u and v into expansions
of these quantities. In a later paper Tsuchiva has also dealt with the effects of the
inertia and frictional terms but these do not seem to alter the previous results. In the
immediate vicinity of the equator the east-west velocity component of the current
is determined by the curvature of the isobaric surface in the meridional vertical
section and not by the slope. The geostrophic approximation for the ocean currents
can be used much closer to the equator than has so far been done. The method used
by Tsuchiva is purely mathematical and not founded on any physical basis.

Cliapter XVIII

Basic Principles of the General Oceanic

Circulation

1. Introduction

The ultimate cause of all movements in the sea is the supply of energy by solar
radiation. The meridional variations in the energy supplied lead to regional differences
in the structure of the oceans. The oceanic circulation modifies, however, the distri-
bution of temperature and salinity, which are basically determined by the climate,
and also affects the distribution of dissolved gases in the sea; it therefore has an
indirect influence on the distribution and accumulation of marine life. The general
oceanic circulation is therefore the fundamental problem of oceanography.

The transformation of solar radiation into heat in atmosphere and sea takes place
mainly in the layers close to the interface between air and land, between air and water,
respectively. Other important influences from the hydrosphere on the atmosphere and
the reverse are also localized at the sea surface and in this way the sea surface becomes
one of the most important interfaces of the earth; it is the starting point of both the
atmospheric and the oceanic circulation. The principal factors involved in these, such
as the solar and sky radiation, outgoing radiation, evaporation, precipitation,
melting of ice and the wind stress on the water exert their major effects here. In com-
paring the atmospheric and oceanic circulation the special circumstance should be
kept in mind that the interface (sea surface) which is decisive for the initiation of
vertical motions is situated below the atmosphere but above the sea. Therefore, in
order to start a vertical circulation in the atmosphere air must be lighter than the
surrounding air masses (rising motion), while in the ocean water as compared with
the surrounding waters must be denser (sinking motion). The variable position of
this interface, from which the vertical circulations originate, causes corresponding
differences of the circulation system (Defant, 1929).

According to the general causes, mentioned above, of steady water movements in
the sea, two fundamental factors stand in question:

(1) the internal field of force of the mass structure, and

(2) the external field of force due to the winds.

Other less important external forces such as the supply of water by precipita-
tion or its removal by evaporation are less effective than the wind forces (see
p. 572).

These two basic factors act quite differently on the water movements and an under-
standing of the general circulation can only be based on the resultant of the two
effects. Most investigations have been limited to the components of motion of the

556

Basic Principles of the General Oceanic Circulation 557

circulation in a meridional plane with only supplementary extensions to three-
dimensional space. This has no doubt been unavoidable in the past due to the lack of
sufficient observations, but a complete understanding of the oceanic circulation can
be obtained only in terms of spatial phenomena. The magnitude and the complexity of
the problems makes it understandable that a solution in full detail has not yet been
obtained and probably will not in the near future, but the accumulation of further
data and the advance of theoretical knowledge will lead closer to a comprehensive
elucidation of the mechanism of the general oceanic circulation which is the aim of
oceanography.

The permanent oceanic currents can be divided into three groups according to their
genetic origin:

(1) currents produced by thermo-haline convection, mainly due to cooling of
surface water in higher latitudes;

(2) currents produced and maintained by the transfer of wind energy to the sea
surface ;

(3) currents maintained by the excess of precipitation over evaporation, or vice
versa occurring in special oceanic regions.

Each of these types of flow shows a different physical behaviour and acquires on
the rotating earth an individual form, which is also strongly influenced by continental
slopes acting as barriers for the oceanic movements.

2. Oceanic Sea Surface Currents

(a) Charts of Sea Surface Currents

It has taken quite a long time until data on sea surface currents were that numerous
as to allow a reliable representation of the currents over the entire ocean surface.
Charts of currents presented in ordinary atlases are seldomly based on critically
tested observations and are often constructed making hypothetical assumptions. As
amount and density of the observational material (current measurements) increased,
charts of current conditions over smaller oceanic areas could gradually be extended
until finally world maps of ocean currents could be constructed. At the suggestion
of Neumayers (1898), Schott prepared a world chart of ocean currents. A new edition
of this was published in 1942 incorporating in an excellent manner the oceanographic
progress of the last 40 years. This chart (Schott, 1942), Deutsche Admiralitatskarte
no. 1947, 2 sheets, 1942) shows the total earth for the Northern Hemisphere winter
and an inset map for 30° N. to 20° S. shows seasonal variations for the tropics during
the Northern Hemisphere summer. North of 50° N. the chart represents more summer
conditions for which the data are more numerous. This current chart is reproduced in
Plate 8 on an equal area projection. The use of current arrows has been simplified in
places: velocities are indicated at \ knot intervals with a lower limit of 12 nautical
miles in 24 h and an upper limit of 36 nautical miles in 24 h. Differences in velocity are
indicated by the thickness of the arrows and the constancy of the current by the length ;
the last factor was expressed in four degrees : variable, fairly steady, steady and very
steady corresponding roughly to 25, 25-50, 50-75 and 75% flow displacement in the
direction of the arrow. Naturally in such large-scale charts only a somewhat general
representation of the currents can be given and some subjective interpretation is
always possible. Details in the infrequently navigated parts of the ocean are, of course,

558 Basic Principles of the General Oceanic Circulation

highly deficient and must be supported by theoretical deductions. For details in parti-
cular areas of the ocean, reference must be made to special charts; the literature
sources will be indicated below.

As is apparent from the current charts in Plate 8, the more schematic distribution of
oceanic currents known from earlier work is really present to a large extent in all
oceans. Northern and southern equatorial currents characterize everywhere the
tropical surface circulation and are usually separated by an equatorial counter current
flowing in the opposite direction, while the surface circulation of higher latitudes is
composed principally by the West Wind Drift and the Polar Current. Separation of
these current regions gives convergence and divergence lines which are specially
indicated in the current chart. They are rarely clear-cut lines; instead they are usually
rather wide areas intruding between individual currents. It is often difficult to deter-
mine their position accurately since they move backward and forward periodically
in time. The connection of this surface current system with the currents of the deeper
layers lies in these singularity areas, and they are thus of great importance.

In the following sections a brief description will be given of the surface-current
conditions in the individual oceans and of their seasonal variations. The dynamics of
single currents will be dealt with later.

{b) The Surface Currents of the Atlantic Ocean

The backbone of the system of currents present in the Atlantic is formed by the two
equatorial currents; that in the Southern Hemisphere is the stronger one and is more
constant and of greater extent. During the whole of the year this current crosses
the equator from west of the island of St Thome until the South American coast.
The meridional distribution of the current intensity shows a double current core for
nearly all months; one of the two just north of the equator at about 1° to 2° N. and the
other one at about 4° to 5° S. (especially between 20° to 30° W.). Between them along
the equator is the equatorial region of divergence which belongs to the tropospheric
deep sea circulation (p. 595). This divergence coincides with the tongues or island of
cooler water that are shown in temperature charts, particularly in the period from
June to August and indicate the upwelling of deep water accompanying the diver-
gence. In the central part (8° to 40° S.) the South Equatorial Current is most intense
from June to July and hardly drops below 20 nautical miles in 24 h. The southern
current core divides into two parts at Cape San Roque — one turning south and be-
coming the Brazil Current, and the other joining the northern current core in the
latitude of the Amazon estuary to form the strong Guiana Current flowing along the
South American coast.

The Northern Equatorial Current is less constant in extent and strength. Its northern
boundaries fluctuate, but from about 20° N. its itensity decreases and it passes into
an extensive region of weak and variable currents with frequent motionless areas.
South.of 20° N. its average intensity is about 15-17 nautical miles in 24 h. Schumacher's
monthly charts (1940) which give greater detail show the eff"ect of the bottom topo-
graphy on the current system where it passes over the mid-Atlantic Ridge (see p. 435).

During the winter months when the equatorial counter current is very weak the
North and South Equatorial Currents flow together along a convergence line from
about 20° W., 4° N. to approximately 50° W., 11° N. but during the summer months

Basic Principles of the General Oceanic Circulation

559

when the counter current is more strongly developed this only occurs between 50° W.,
10° N. and 60° W., 14° N. From here a combined current runs in a westerly direction
towards the West Indies throughout the whole year; this is the source for the surface
currents in the West Indies and therefore also for the Gulf Stream (Dietrich, 1937 b;
1939), which is in agreement with the results of Brooks (1930, see also, Shaw and
Hepwort, 1910) showing that the fluctuations in the south-east trade winds are more
closely connected with water and air temperatures in Western Europe than are those
of the north-east trade winds.

The Equatorial Counter Current lies between the two equatorial currents. Table 147
presents its position in different seasons. During almost the whole of the year it is
divided into two parts; the "western" counter current weak and not very broad,
found particularly during the first winter months and the "eastern" counter
current which is present all the year round. Only in the summer months do they
join, thereby forming a mighty counter current. The origin of this lies west of 50° W.,
near the American coast, its width covers the area between 10° and 3° N. showing
considerable speed and constancy. During the period of its greatest extent the central
area of the current is characterized by a convergence region towards which water
flows from both sides. An attempt has been made by Schumacher (1940) to show
a connection between the temporary interruptions in the counter current above the
mid-Atlantic Ridge and the topography of the rise.

Table 147. Extent of the Equatorial Counter Current in the Atlantic Ocean
(according to Schumacher)

Region with

Western Counter Current

Eastern Counter Current

no currents

(deg. lat.)

January 53' W.

10° N.

until

37° W.

, 6°N.

26°W., 7°N. ^
19° 5°

February 49°

until

41°

6°

March 53°

10°

until

47°

7°

20° 4°

April 52°

until

37°

0°

24° 4°

May 47°

6°

until

33°

QO**

28° 5°

June 51°

9°

until

38°

3°**

36° 5°

until the

July 51°
August 56°

90
10°*

—

^African
coast

0
0

September 52°

10°*

—

October 53°

10°*

—

November 54°

10°*

until

32°

8°

31° 7°

December 51°

until

30°

6°

29° 6° J

* Starts presumably farther north-west,** with interruptions.

Northern Hemisphere. The combined equatorial currents enter the Caribbean Sea
between the Antilles and spread over almost its entire width as the Caribbean Current;
this flows almost due west with its greatest velocities in the southern part. In some
months large vortices are formed off" the coast of Costa Rica, Panama and Colombia.

560

Basic Principles of the General Oceanic Circulation

100°

20 -

100

20 -

!00

Fig. 256. Schematic picture of the sea surface currents in the Gulf of Mexico (according

to Schumacher).

The current then enters the Gulf of Mexico through the Yucatan Channel with veloci-
ties of up to 3-7 knots at the current core. The currents of this mediterranean sea are
shown in Fig. 256 (Schumacher, 1940). The major part of the stream lines leaving the
Yucatan Strait tend to circle or cross the Gulf clockwise following the shelf line. The
branch that flows directly to the Florida Straits is stronger and is steady only during
the winter months.

The eastern branch of the Yucatan Current forms the Florida Current the water
transport of which is the main source of the Gulf Stream. No other ocean current has
been so intensively investigated as this. An enormous amount of literature has been
accumulated on the subject that is impossible to cite here in detail. The water piled
up in the Gulf of Mexico flows out through the Florida Straits towards the north as a
gradient current (Florida Current) against the prevailing winds. This current becomes
stronger where the channel narrows off Bimini and may have a velocity of over 60

Basic Principles of the General Oceanic Circulation 561

nautical miles in 24 h with up to 80-100 nautical miles in the current core. These
values correspond to about 1 •5-2-5 m/sec which is hardly reached even in the down-
stream parts of big rivers. According to Krummel (1911, p. 576), the axis of the stream
under steady conditions is:

35 nautical miles in the Yucatan Channel (east of Contoy Island),

25 nautical miles north of Havana (85° W.),

11 nautical miles east of Fowey Rocks (Florida 25-7° N.),

19 nautical miles east of the Jupiter light tower (Florida 27° N.),

38 nautical miles south-east of Cape Hatteras.

At the edges, particularly on the western side, the current shows often variations in
direction and strength. Not infrequently there is a counter current flowing in a south-
westerly or westerly direction along the Florida Keys into the Gulf of Mexico and is
well separated from the basic Gulf Stream. It is connected with the counter current
always found further north off the east coast of America. In the most narrow parts
of the channel the current has a width of about 30 nautical miles, off Cape Canaveral
(28-5° N.) about 60 and off Charleston a width of as much as 120 to 150 nautical
miles. In general, the western border of the blue coloured warm water of the current
follows the continental slope. To the west of it on the shelf the cold green water of the
"cold wall" is usually travelling slowly to the south; (see Pt. I, p. 144, Fig. 60). The
Florida Current is joined here by the important Antilles Current flowing north-west
to the north of the Bahamas. Before the junction (27° N.) it is narrowed in the con-
vergence region of the Sargasso Sea, whereby it becomes of some importance (see
Nielsen, 1925; Wiisx, 924). North of Cape Hatteras the Gulf Stream turns farther
and farther away from the continental slope, possibly due to offshore winds, Coriolis
influence and the increasingly strong cold coastal current of low salinity. This is the
beginning of the second part of the Gulf Stream. Its left-hand boundary remains
sharply separated from the coastal waters but the right-hand edge is extremely blurred.
Here, due to the deflection of the stream lines a counter current is formed which,
although narrow, weak and variable is a characteristic phenomenon of the eastern
flank of the main current, but because of its narrowness it can rarely be detected by
means of ship displacements ; however, the farther to the north-east the stronger and
more frequent this current appears. Only mean positions of the current can be deduced
by evaluation of the average physical conditions at the sea surface. Better results can
be obtained by systematic recordings of the sea-surface temperature at short time
intervals ; these then give a more accurate indication of the mean position of the warm
Gulf Stream core and also of its northern and southern limit (see Pt. I, p. 144, also
FuGLiSTER, 1947). Determinations of the Gulf Stream position obtained by different
methods can be combined to give an average picture (Neumann and Schumacher,
1944) but it should always be borne in mind that the boundaries of the warm- water
belt cannot necessarily be regarded as identical with the boundaries of the current.

From about 55° W. the left side of the Gulf Stream is flanked by the cold and weakly
saline water of the Labrador Current. At this polar front the cold water masses sink
below those of the Gulf Stream and thereby numerous vortices are formed. To the
south of the Newfoundland Banks the Gulf Stream turns sharply towards the south
(p. 421) and again back towards north and from here gradually widens and splits into

562

Basic Principles of the General Oceanic Circulation

current branches of varying strength and of varying temperature. From Cape Hatteras
to the Irish coast its direction remains mainly eastwards or north-eastwards ; the average
velocity falls from 15 to 5 nautical miles in 24 h and its constancy from 70 to 30%.
The almost synoptic surveys of the International Gulf Stream Expedition of 1938
showed that the Gulf Stream to the north of the Azores is no longer a single current,
but is broken up into several branches flowing to the north-east as warm and highly
saline intrusions between cold, weakly saline water masses moving slowly in the
opposite direction. Neumann (1940) has shown that this finger-like ineraction of

38° W

36°

32°

26"

48°

48»

46°

•7

.44'

f.m

]■

42°

Jl.

t^=^ S^::^

40"

^'^

i:^

42°

f -^ rr\rM-P.S 0 „ . ^-^

F\^

40°

olOO J

Azores

o^ ^

38°

36<

38° yy 36°

28°

Fig. 257. Most probable course of the Gulf Stream north of the Azores in June 1938.
(The open arrows indicate the assumed position of the cores of individual branches of

Gulf Stream.)

Basic Principles of the General Oceanic Circulation 563

different water types was no chance phenomenon present in June 1938 but is a per-
manent feature of the current in these regions (see Fig. 257).

In the eastern half of the ocean the Atlantic Current divides into two main branches
at about 20° W. ; one of these flows north-east past Ireland and with a reduced strength
and moderate Constance through the Faeroes — Shetland Channel into the Norwegian
Sea and along the Norwegian coast. It is still noticeable in the Arctic Ocean. The weak
and variable second branch turns east-south-east towards the French and Spanish
coasts (the Portugal Current). The stronger and also more steady Canaries current in
the south-eastern North Atlantic cannot be regarded as a continuation of the Gulf
Stream (Thorade, 1928). It seems to be advisable to refer to the whole current from
the Florida Straits to the Norwegian coast as the Gulf Stream System but to distin-
guish six separate parts of this system (Iselin, 1938); the most important are:

(1) the Gulf Stream close to the coast or the Florida Current (from the Gulf of
Mexico to Cape Hatteras) ;

(2) the Gulf Stream in the open ocean (from Cape Hatteras until north of the
Azores) ;

(3) the Irish Current (from the splitting point until the Faeroes — Shetland sill);

(4) the Atlantic (or Norwegian) Current (along the Norwegian coast).

A side branch of the Irish Current flowing from the south of Iceland to its conver-
gence with the East Greenland Current is called the Irminger Current. Helland-
Hansen and Nansen (1909) deduced the sea surface currents of the Norwegian Sea
from an analysis of temperature and salinity in charts and vertical sections (Fig. 157,
p. 368). North of the Lofoten the Atlantic current divides into a branch flowing
towards north and north-west (towards Spitzbergen) and another one flowing north-
east into the Barents Sea (Schulz, 1929). Towards Greenland the East Greenland
Current is still wide and strong north of the Denmark Strait. In the central part of the
Norwegian Sea there is an extensive area of extended vortices apparently connected
with the topography of the sea bottom.

Southern Hemisphere. The Brazil Current is a continuation of the South Equatorial
Current from Cape San Roque southward. Between 15° S. and 20° S. it is still inside the
region of the south trade winds. Off" Cape Sao Thome and Cape Frio the main current
flowing south-westwards shows a contraction from its eastern (left) side during most
months ; from here it follows the continental shelf line fairly close, probably due to the
influence of the Coriolis force. Over the shelf a counter current exists which can be
regarded as a branch of the current along the Patagonian shelf (Falkland Current).
Off the La Plata estuary the eastern part of the Brazil current turns south-eastwards
working into each other in a finger-like fashion with the Falkland Current flowing
from the south-west. Near the coast the Falkland Current intrudes to the north and
north-east as far as 35° S., deflecting the Brazil Current to the east. Between the two
opposing currents there is thus a sharp convergence line formed which is clearly shown
by the distribution of the oceanographic factors. This gives rise to vortices found in
this part of the ocean. The interaction between Falkland and Brazil Current form a
southern hemisphere counterpart to the Labrador Gulf Stream system in the Northern
Hemisphere, but the first ones are less well developed and of less intensity.

The area of the West Wind Drift includes the whole of the southern part of the
South Atlantic Ocean between about 35° and 63° S. It belongs to the large circumpolar

564

Basic Principles of the General Oceanic Circulation

i;0° 100° 90° 80° 70° 60° 50'

50° 40° 30° 20° 10° 0° 10° 20° 30* 40° 3U-

\W 120°

Kk)° 90° 80° 70° 60 50° 40° 30° 20° 10° 0° 10° 20° 30° 40^

60° E

Fig 258 Singular lines in the current field of the sea surface in the Atlantic Ocean.
(A) 'in the system of the tropospheric circulation: (1) the divergence region m the area of
the Cap Verde Islands (7° to 15^ N.); (2) the equatorial divergence region; (3) the con-
vergence region in the Equatorial Counter Current. In the region of the tropica thermoclme
these singular lines correspond to inverse ones. (B) the divergence region of the Benguela

Current (C) , subtropical convergence; , polar and equatorial limits of the

subtropical convergence regions. (D) , the oceanic polar front (Arctic and Ant-

arctic convergence).

Basic Principles of the General Oceanic Circulation 565

current which keeps the water masses constantly in motion around the earth from west
to east. It is of much greater strength and constancy than the corresponding West
Wind Drift in the North Atlantic. South of 35° S. and east of 20° W. it flows mainly
in a north-easterly direction. There are widely differing opinions about the position of
its northern boundary in the area of the subtropical convergence; the southern
boundary is found at about 63° S. but is not sharply defined either. At the core of the
West Wind Drift lies the boundary between two quite different water types, the
subantarctic water of middle latitudes and the Antarctic polar water. In the Atlantic
this latter water type has its origin almost entirely in the Weddell Sea. A small part
only comes from the Pacific through the Drake passage. The boundary between the two
water bodies is denoted the South Polar Front {Antarctic Convergence) on both sides
of which the currents flow between east and east-north-east but the velocity is greater
on the northern side. For the dynamics of this front see p. 549.

The Polar Current in the Southern Hemisphere flows in the coastal regions of the
Antarctic carrying cold polar water westward until the Weddell Sea where it turns in a
great arc around a central almost motionless region and flows towards north or north-
east to become the southern part of the West Wind Drift. East of 10° W. the course of
this Antarctic polar current coincides almost entirely with the mean pack-ice limit of
the southern summer.

The framework of the circulation system of the sea surface formed by singular
lines and regions inside the current field is shown in Fig. 258. In the tropical and
subtropical circulation the divergence lines stand out clearly in the eastern parts of the
North and South Equatorial Currents. In almost all months there is a narrow area of
divergence off the West African coast in particular between the Canaries and the Cape
Verde Islands that extends towards the south-west beyond 35° W. as a two-sided
divergence line and forms the southern boundary of the North Equatorial Current.
This is connected with the upwelling of cold water off the West African coast. Its
counterpart in the Southern Hemisphere is the extended divergence line in the area
of the Benquela Current off the coast of South West Africa; the upwelling of cold
water also occurs here (Defant, 1936a). Reference has already been made to the
divergence line along the equator between the northern and southern branches of the
Equatorial Current (p. 559) and also to the convergence line in the Equatorial Counter
Current. The Cape Verde divergence line, the equatorial divergence line and the con-
vergence line that lies between them are all part of the tropospheric circulation system
and are associated with contrary singularities in the lower layers of the troposphere
(p. 595).

The oceanic regions between the Equatorial Currents and the West Wind Drifts
in both hemispheres contain weak and variable currents. Stream lines deflected to the
right from the Atlantic Current and from the North Equatorial Current together form
the region of subtropic convergence. This extends across the Atlantic from 75° to 20° W.
but is not a continuous uniform convergence line. Vortex formations are the charac-
teristic type of motion with the existing slight density differences. In these vortices
warm water sinks to become part of the warm-water mass of the troposphere in this
region. This convergence is always indistinct and shows everywhere large seasonal
variations (Felber, 1934) and is therefore more appropriately called a subtropical
convergence region than a convergence line. In this convergence region the interaction

566 Basic Principles of the General Oceanic Circulation

between highly saline and warm water from lower latitudes with weakly saline and
colder weater from higher latitudes lead to vortical movements of large extent. Similar
conditions are found in the subtropical convergence region of the South Atlantic.
There are rather different opinions about the question how far the West Wind Drift
reaches equatoward depending on whether the subtropical convergence is fixed
according to ship displacements or if it is derived by means of the distribution of
oceanographic factors. The position given by Deacon (1937), deduced mainly from
the temperature distribution, is always about 6° to 10° further south than that obtained
from current measurements. According to Bohnecke (1938, p. 201) the "subtropical
convergence" (of the currents) should be carefully distinguished from the "subtropical
boundary" (deduced from temperature and salinity). The former in a rather charac-
teristic way coincides with the tropic boundary and the latter with the polar boundary
of that large disturbance region which extends between the southern limit of the
Equatorial Current and the West Wind Drift (p. 564) as is found during the dynamic
preparation of serial observations. Also here it seems more appropriate to speak of a
convergence "region" between the two bordering water types being the place for
subtropical vortex formations.

The Southern Hemisphere Polar Front (Antarctic convergence line) has been dis-
cussed on p. 549. The Northern Hemisphere Polar Front is sharply developed between
the Labrador Current and the Gulf Stream near the Newfoundland Banks but
gradually fades towards the north-east, reappearing again as a frontal zone between
the East Greenland Current and the Irminger Current. Larger and smaller vortex
formations with corresponding vertical movements are also found along this con-
vergence line.

Ships displacements available for other oceans are much less numerous than in
most parts of the Atlantic and current charts are therefore correspondingly more
uncertain. Reference to analogous conditions as in the Atlantic will usually permit
briefer description here, but the Indian Ocean has a single particular peculiarity in its
northern part where the wind system changes character completely every six months,
correspondingly causing similar changes of the ocean currents. This is the best
possible proof that the winds are decisive for the generation and maintenance of ocean
currents. A full cartographic description of the currents here requires monthly charts
(British Admiralty 1895; Deutsche Seewarte 1908; Dallas and Walker, 1908;
MoLLER, 1929) but charts for the summer monsoon and for winter are usually con-
sidered sufficient.

The currents during the time of the north-east monsoon (north-east trades) corres-
pond best to the general system of ocean currents. They resemble those of the Atlantic
and the Pacific except that the Equatorial Counter Current lies between about T S. and
8° S., that the Northern Equatorial Current moves partially into the Southern
Hemisphere; during this part of the year the thermal equator is always south
of the equator. In the north the North Equatorial Current (monsoon drift) runs
almost due west. It is strongest to the south and south-west of Ceylon where the cross-
section through the current is narrow. In the Bay of Bengal there is an anticyclonic
vortex. The strong north-west to north-east winds over the Arabian Sea produce a

Basic Principles of the General Oceanic Circulation 567

drift current towards west-south-west or west. Thereby a current boundary is formed
beginning north-west of Cape Comorin and can be followed along about 10° N.
westwards until 60° E. It carries the character of a convergence line between water
from the Arabian Sea and water masses of the main current flowing from the east.
ScHOTT (1928fl) has mentioned the great contracts in surface salinity here. Part of this
water transport into this region enters as a very strong current into the Gulf of Aden
and continues through the Strait of Bab el Mandeb into the Red Sea. The other part
forms a strong south-west current flowing along the Somali coast to about 7° S., where
the Equatorial Counter Current starts rather abruptly having a direction towards east.

South of the counter current flows the broad South Equatorial Current and shows
large seasonal variations in velocity and constancy caused by the annual variation of
the south-east trade winds. The current core lies near the northern boundary of the
current at about 10° S. to 15° S. in both summer and winter (Michaelis, 1923). The
irregularities in the South Equatorial Current due to Madagascar have been investi-
gated by Paech (1926). In the Southern Hemisphere summer a "Stau" current flows as
a southward current along the African coast starting at 10° S., the Mozambique
Current, with a tributary current from the east coast of Madagascar. Both form the
source for the Agulhas Current at about 30° S., which continues closely to the conti-
nental shelf until it swings out from the shelf around the Agulhas Bank at the southern
tip of Africa. The northern part of the core, however, still keeps to a very large extent
over the contmental shelf. From the southern end of the Agulhas Bank part of the
current then flows north-west as the Benguela Current and part turns back into the
Indian Ocean forming a series of large vortices. The complicated nature of the
currents in this part of the convergence zone between the Agulhas Current and the
west wind drift is clearly shown in an analysis of the current field which has been
prepared by Merz (1925).

The atmospheric pressure and wind distribution over the Indian Ocean north of the
equator changes drastically during April. Almost immediately the sea surface currents
react to this change in the wind direction and at the same time there is a redistribution
of the water piled up at the coasts. The South Equatorial Current still remains in the
Southern Hemisphere (south of 5° S.) but is considerably intensified. The counter
current disappears and over the entire northern part of the ocean except the coastal
zones a fairly constant eastward current appears, the South-west Monsoon Current.
The convergence line between the South Equatorial Current and this monsoon current
is well developed along the total width of the ocean and broken only in the extreme
west where a strong branch turns northwards from the South Equatorial Current
between 5° S. and 0° and flows along the coast into the Arabian Gulf as the Somali
Current. It follows closely the steep pressure gradient off" the coast between the region
of piled up water ("Anstau"-Gebiet) between 5° and 10° S. and the area from which
water has been removed by the monsoon current between 5° N. and 10° N. This is
accompanied by upwelling just off" the African and Arabian coasts (Puff, 1890). The
Somali Current possesses mostly an extreme intensity, so that speeds here are greater
than in the Florida Current (often more than 100 nautical miles in 24 h) (Fig. 259).
The formation of anticyclonic vortices to the south-east of Ras Hafun and the marked
concentration of the current core into a narrow coastal belt is characteristic and
accords with the increase of the Coriohs force towards north.

568

Basic Principles of the General Oceanic Circulation

55°

36 ^i^ h^^J C^^^<^,^^^J^U.^^I^

'^42 ,3p>t. \ 22-^ "-^v/ ^ O

3Jr- ,-i~

\CP lol -"^

-.._--V^

7 18 . „,

6?^

Fig. 259. Current displacements in the Somali Current at the time of south-west monsoon.

The southern boundary between the current branches of the South Equatorial
Current and the West Wind Drift is again a long convergence line at about 40° S.
For its position see Willimzik (1929) and the alternative interpretation by Schott
(1925, p. 163). South of the convergence region and especially in higher latitudes the
West Wind Drift has a very low constancy corresponding to the variable winds of this
region. The non-uniform character in the current is already shown by the rapid decrease
in constancy as the number of observations increases. The Antarctic Comergence
runs right across this broad current gradually receding from 48° S. in the west to
about 54° S. In this area the West Wind Drift flowing east-south-east meets the cold
coastal Antarctic water flowing west-north-west and north-west (Willimzik, 1927).

(d) Sea Surface Currents in the Pacific Ocean

The principal currents of the Pacific are again the North and South Equatorial
Current. Because of the great width of the Pacific they are almost purely east-west

Basic Principles of the General Oceanic Circulation 569

currents. Since the thermal equator remains in the Northern Hemisphere throughout
the whole year these currents are not symmetrical about the geographical equator.
The southern boundary of the North Equatorial Current lies between 6° N. and 7° N.
in winter and between about 9° N and 11 ° N. in summer. It is much stronger in winter.
At its southern boundary the current at each location has a purely zonal direction and
constant speed, while its velocity increases steadily towards the west. Off the Philli-
pines (north of Mindanao) the strong current divides : one branch flowing northward
to become the Kuroshio and the other turning sharply southward into the Equatorial
Counter Current. Off the east coast of Mindanao it flows southwards with a 100%
constancy (Schott, 1939, see also Puls, 1895).

The South Equatorial Current covers the wide south-east trade wind belt between
about 5° N. and 40° S. The greatest velocities and constancy again lie along the
northern border between 5° N. and 5° S. and, as in the Atlantic, a double current core
is occasionally present. By this a long and narrow tongue of extremely low tempera-
ture is caused in the thermal field in the eastern part of the Pacific west of the Gala-
pagos Islands. These areas of cold water are associated with the occurrence of
eastward ship's displacements within the South Equatorial Current. Similar ship's
displacements are occasionally observed in the Atlantic. West of New Guinea and the
Solomons the South Equatorial Current during the northern summer is a torrent
current extending almost as far as Halmahera ; it supplies the main water mass of the
counter current. Off the east coast of Australia the South Equatorial Current bends
and is called from thereon the East Australian Current which corresponds to the
Agulhas Current in the Indian Ocean.

All the year long a well-developed counter current is inserted between the two
Equatorial Currents. During the northern winter it is weak and narrow, except in its
starting area in the west, but during the northern summer especially during August
and September it flows with great Constance from Mindanao-Palau-Halmahera to
Panama (almost 8000 nautical miles) with a width of about 300 miles between
5° N. and 10° N. It is separated from the Equatorial Currents by well-defined bound-
aries especially on the northern side.

The Kuroshio is a continuation of the North Equatorial Current and in many
respects an important phenomenon for Eastern Asia. A review of what is known of
this current and a comparison with the Gulf Stream system with numerous references
has been given by WiJST (1936^, see also, Uda and Okamoto 1930, 1931 ; Uda, 1933).
In summer it starts flowing northward east of Formosa with a velocity of 24-36
nautical miles in 24 h and a width of about 300 nautical miles. Then it runs west of
the Ryukyu Islands between the Ryukyu Ridge and the East China shelf with decreas-
ing width and correspondingly increasing speed (36-48 nautical miles in 24 h) until it
branches south of Japan ; one branch, the Tsusima current enters the Sea of Japan and
flows north-north-west, the other, the proper Kuroshio, flows with a reduced width
along the south-eastern coast of Japan. Between 31 ° and 35° N. it is only about 150 km
wide but its velocity rises to 48-56 nautical miles per day. Its left-hand boundary is
sharply defined but the right-hand one (oceanic side) is blurred. Here, like the Gulf
Stream, it has a weak counter current. It turns abruptly eastwards towards the open
ocean at 36° N. off the Boso Peninsula with an almost invariable width but with
gradually decreasing velocity (48-24 nautical miles per day). This deflection of the

570

Basic Principles of the General Oceanic Circulation

current has been regarded by Hidaka (1927-28) from experimental evidence as due
to the change in direction of the north-east coast of Japan, but Wiist believed that
topographical factors south of the Boso Peninsula were responsible. The Kuroshio
extends out into the open ocean as a relatively strong current along 34-36° N. as far
as 175° E. a distance of about 1,600 miles. Only for a short distance along the coast
the current keeps the north-east direction. Figure 260 shows a schematic representation
of the main current cores of the Kuroshio system during the summer as given by
Wiist. Table 148 gives a comparison with the Gulf Stream system.

Table 148. Comparison between Kuroshio and Gulf Stream
(Mean values for summer, according to Wiist)

Current
section

Width
(km)

Direction

Speed
(cm/sec)

Nautical
miles/24 h

Temp.
(°C)

Salinity

/oo

Kuroshio

23°-24°N.
27°-28° N.
31°-33°N.
About 36° N.t .

300
230
150*
150

N. to E.

N.E.
N.E.
E.

51-77

77-103

100-120

51-100

24-36
36-48
48-56
24-48

5^22-28

34-8-34-9

Gulf Stream

23°-24°N.
27°-28°N.
31°-33°N.
About 36° N. .

110
140
180:
180

N.
N.E.toN.

N.E.

100-120

140-160

About 120

About 100

48-66

66-75

About 56

About 48

I 22-28

360-36-4

* After separation of the Tsusima Current.

t 400 km east of the Japanese Coast.

1 After confluence with the Antilles Current.

The Oyashio flows south-west to south-south-west in the dead angle between the
north-west coast of Japan and the north-western branch of the Kuroshio as far as
37° N. It is a relatively cold current with a very low salinity (33-5"/oo). It does not reach
as far south in summer as in winter. According to Uda the boundary between the
Kuroshio and the Oyashio as a convergence region consists of numerous vortices
similar as at the boundary between Labrador Current and Gulf Stream. Differences
in temperature and salinity across this convergence line in winter will be at least as
large as those off the Newfoundland Banks. Driven by the strong northerly and
northwesterly winds the Oyashio takes its cold water supply from the Sea of Okhotsk
near the Kuriles and in part also from the Bering Sea.

The water is mostly in slow motion between the cold boundary which runs east-
wards a little north of 40° N. and gradually fades away and the subtropical conver-
gence which begins in the west at 20° N. and turns northward, at first only slowly,
to reach 35° N., remaining in this latitude until about 138° W. This continuation of
Kuroshio is termed the North Pacific Current. It main part turns southward between
150° and 135° W., part joining the California Current and part mixing with the water
from the North Equatorial Current along the subtropical convergence.

120

1%0°

20'u

— -- 200m„ — — ,IOOOm_-._~..2000m
Mansyu stations May- June I925-?
Monsyu stations Jan. Feb 1927

lao*

'20°

130°

^HQ' E

Fig. 260. Main current branches of the Kuroshio system (according to WUst). ( 1 ) Kuroshio

(main current); (2) Tsusima Current; (3) Korean side-branch; (4) northern branch of

Kuroshio; (5) Oyashio; (6) Liman Current; (7) Counter currents of the Kuroshio. At R

position of the Riu-Kiu section, at S position of the Shiono-Misaki section.

Basic Principles of the General Oceanic Circulation

571

The northern part of the North Pacific Current turns northward and flows in an
anticlockwise direction around the Gulf of Alaska; it is a well-developed current and
is fairly constant, particularly near to the coast. This Alaska Current flows along the
Aleutians and extends into the southern Bering Sea through all the passages between
the islands. In the eastern part of the Pacific the southward movement off the Cali-
fornia coast is denoted the Californian Current (Thorade, 1909; Warmer, 1926).
It replaces the water which is carried westward by the north-east trade winds. The
north-east to south-west direction of the current indicates the presence of an ofi"-
shore movement, giving rise to the upwelling of cold water along the greater part of
the Californian coast. This upwelling occurs mainly during the warm part of the year.

The northward to north-westward movement of water along the entire western
coast of South America is called the Humboldt Current after its early investigator.
Where it runs close to the Chilean and Peruvian coasts it is called the Peru Current
and this current and its variations have been described in a detailed monograph by
ScHOTT (1931). A later evaluation of the available data has been given by Gunther
(1936, 1936a). Figure 261 shows the probable field of motion according to Schott for the
two seasonal extremes. During the period of intensified trade winds in the Southern
Hemisphere winter (Chart a, Aug.-Sept.) the Humboldt drift current and its con-
tinuation, the South Equatorial Current, intensify considerably. The strength of the
current rises from 0-5 to 0-7 knots along the coast of northern Chile and Peru and
increases to 1 and occasionally 2 knots where it flows north-westwards in a wide
region around the Galapagos Islands. Further out to sea it turns westwards. The

W-Lq

W-Lg

Fig. 261. Most probable current pattern in the region of the Humboldt Current and north

of it (according to Schott): {a) for the Southern Hemisphere winter (August/September);

(6) for the Southern Hemisphere summer (February/March).

572 Basic Principles of the General Oceanic Circulation

coast as far as 5° S. is thus a one-sided convergence line and as a consequence up-
welling occurs along its entire length. The other extreme of seasonal variation is at
the end of the Southern Hemisphere summer (Chart b; Feb.-Mar.). Conditions in the
equatorial region at this period are very complex and unstable and are subject to the
influence the more or less pronounced development of the Equatorial Counter
Current and the North Equatorial Current. The Humboldt Current is now weaker
and about 4° C warmer at the coast. The unstable character of the current is due to
simultaneous instability in meteorological conditions in the entire area between the
Cocos Islands, the Galapagos and the coast of Ecuador and Peru. In many years the
thermal equator and the associated zone of minimum atmospheric pressure are dis-
placed into the Southern Hemisphere, so that the south-eastern trades along the
Peruvian coast are then disturbed and rainy north and north-west winds occur in
northern Peru. These disturbances of atmospheric and oceanic conditions are, how-
ever, usually not too powerful, but in general conditions are so unstable in northern
Peru that abnormal developments frequently occur. The warm weakly saline water of
the Equatorial County Current can then easily advance into the area of the Humboldt
Current. This warm water is then carried southward by the northern and north-
western winds (most often at Christmas time). This current in contrast to the Peru
Current is regarded as a "counter current"; it is called "El Nino". Normally the
changes are not very great but occasionally when the disturbances are particularly
well developed there may be torrential rains followed by flood catastrophes in coastal
areas of northern Peru which are adapted to a dry climate. The simultaneous change
in the character of the water masses off" the coast in addition has disastrous conse-
quences for the guano birds which are suddenly deprived of food. Detailed descriptions
have been given for years when these disturbances have been particularly well marked,
for 1925 by Zorell (1928); Murphy (1926) and Schott and for 1891 by Schott
(1931).

The wide area of the Pacific covered by the essentially eastwards flowing West
Wind Drift extends south of the subtropical convergence which is more a "con-
vergence region" than a line. The available data on this current, especially in the
thirties and forties, is rather uncertain. Near 40° S., off" the South American coast there
exists a zone of remarkably low salinity (34%o) apparently originating from western
Patagonia (Schott, 1934). Corresponding to this distribution the West Wind Drift
must swing sharply north to north-westward, that is, to the left. The Antarctic Con-
vergence runs through the West Wind Drift at about 55° S. It was encountered in
every profile recorded by the "Discovery" Expedition and is the only convergence
line circling the entire earth in the Antarctic region.

3. Currents Caused by Excess of Precipitation and Run-off Over Evaporation

The possibility of the direct formation of ocean currents due to the flow of excess
water from the precipitation areas and those with run-off" from rivers into evaporation
regions, was first investigated in detail by Ekman (1926) using his classical theory of
deep and bottom currents. For a circular oceanic region he obtained after considerable
simplifications a final equation of the form

277

curl K -^{P- E), (XVIII.l)

Basic Principles of the General Oceanic Circulation 573

where V is the velocity of the deep current produced, p is the average density of the
bottom current layer D and (P — E) is the difference between precipitation and
evaporation in the area under consideration. Estimation of the velocity in some
actual oceanic regions gave maximum velocities of the "evaporation currents" of
not more than 1-2 cm sec"^, but probably only fractions of this value are reached.

This is valid for open sea surfaces. For partly enclosed basins the quantity (P — E)
may be of exceeding consequence ; the current processes occurring with water inter-
change in sea straits have been already discussed before (Chap. XVI, p. 513). Besides
the water transport through the sea straits also the salt transport stand in question. If
the inward water transport is A/,, the outward water transport Mq and the correspond-
ing salt transports are Si and Sq, then under stationary conditions the two equations

MiSi = MoSo and M, = Mo-(P ~ E) (XVIII.2)

are valid, and thus

Mi =(P-E) ^^\ . (XVIII.3)

This formula is identical with the simple Knudsen relations (p. 379). For example,
when the inflow through the Straits of Gibraltar is about 1-75 x 10^ tons sec-\ the
average salinity of the inflowing water about 36-25%o and of the outflowing water
37-75%o, then for the Mediterranean Sea according to the formula (XVIII.3) the quantity
E — P results to 0-07 x 10^ tons sec-\ which is in good agreement with other
estimates.

More recently, Goldsbrough (1933) has dealt with ocean currents produced by
the given distribution of precipitation and evaporation. Already before that Hough
(1897) in his famous theoretical study of tides on a rotating globe has dealt with this
problem of currents produced by a zonal distribution of precipitation and evaporation.
Since he ignored frictional effects, he found a uniformly accelerated system of purely
east-west geostrophic currents as a consequence of these distributions. From the
impossibility of finding a steady state solution he concluded that precipitation and
evaporation cannot be a significant cause of ocean currents. Hough did not accept any
meridional boundaries in the ocean. Goldbrough took instead a model with precipi-
tation predominating in one hemisphere, evaporation in the other and assumed
meridional boundaries in the ocean. This model gave a steady current field, provided
that the integral of the precipitation-evaporation function taken along each parallel
of latitude between the two boundaries, vanishes. This is a very severe restriction which
no natural distribution of precipitation-evaporation necessarily fulfils. Figure 262
shows the current system produced in this case for one hemisphere; the other hemi-
sphere will be the mirror image of this. The field of pressure, the elevation of the free
surface and the flow will be steady. The horizontal velocity components will thus be
entirely geostrophic, and the current will flow along the isobars. The vertical component
will be zero at the bottom and will increase linearly from the bottom up to the sea
surface where it will equal the precipitation-evaporation rate. At the eastern edge of
the precipitation hemisphere there will be two low-pressure cells, and at the western
edge two high-pressure cells. At the poles the flow is directed from the region of
evaporation into the region of precipitation; however, in the opposite direction in

574

Basic Principles of the General Oceanic Circulation

EVAPORATION

PRECIPITATION

Fig. 262. The steady circulation of Goldsbrough type driven by precipitation over one-half
of a hemisphere and evaporation over the other half. Only one hemisphere has been pictured,
for the other hemisphere applies the reflected image. The curved lines with attached arrows
are isobars. The centres of high- and low-pressure cells are to the right resp. To the left of

the middle line.

subtropical and tropical regions. The geostrophic current will everywhere be directed
towards the equator in the precipitation hemisphere and towards the poles in the
evaporation hemisphere. The current towards the equator will require a horizontal
divergence, that towards the poles will require horizontal convergence. This diver-
gence (or convergence) distribution must be suflficient everywhere to absorb (or supply)
the water locally precipitated (or evaporated).

The solution in Fig. 262 is valid for an entire hemisphere but it is evident that a
coastal barrier could be placed along any complete isobar without affecting the solu-
tion. Thus, meridional barriers can be placed tlirough the centres of the precipitation
and evaporation hemispheres, and also the equator itself can be selected as such a
barrier.

This schematic representation of Goldsbrough's results has been discussed here in
some detail, since Stommel has used it as a basis for a discussion of the fundamental
principles of ocean circulation (see Chap. XXI).

4. The Thermo-haline Circulation

The general atmospheric circulation is produced solely by heat differences in
meridional direction, ultimately caused by the sun radiation. By analogy to the

Basic Principles of the General Oceanic Circulation 575

atmospheric conditions it was assumed at an early date that there would be a simple
water circulation in a vertical plane between the equatorial zone and the polar oceans.
This opinion was first expressed by Humboldt (1814, 1845, p. 322) who also offered
a more detailed reason for it. He pointed out that the very low temperatures in the
deeper water layers at low latitudes could only be regarded as a consequence for the
cold water transport in the deeper layers from the poles towards the equator, which
would also imply a surface water transport towards the poles. The entire mass of the
oceans between the equator and the poles including the water at very great depths
would thus be in constant motion, Humboldt considered the differences in density
between equatorial and polar water masses as the cause for this closed circulation
system. Since the circulation is in accordance with the given temperature distribution,
he concluded that the distribution of salinity was not such as to disturb the thermally
produced circulation. Humboldt's ideas were adopted by many investigators and for
three-quarters of a century formed the basis of a generally accepted view on ocean
circulation. Lenz (1847) found that already for small depths, temperatures in the
equatorial regions are much lower than in the subtropics, and he concluded that the
almost horizontally flowing deep current coming from higher latitudes must assume
an upward directed component near the equator. He deduced from this that, sym-
metric to the equator, there must therefore be two major vortices in a vertical plane,
one on either side of the equator with the cold deep currents rising and merging in the
equatorial region; cold deep water would thus be found nearer the sea surface here
than further north or south. He found support for his conclusion in the salinity
minimum of the equatorial zone.

Ferrel (1856) took the Coriolis force into account and proposed a modified form of
Lenz's vortices limited not in the polar regions, but only in middle latitudes, but
followed by another vortex in the polar regions of each hemisphere with a rising
movement near the poles. The analogy between the atmospheric and the oceanic
circulation is particularly evident in Ferrel's model; he completely ignores the differ-
ence due to heating of the ocean from above, and of the atmosphere from below, and
also disregarded the effects of the salinity distribution and winds.

The wide adoption of the thermal circulation theory is due to the circumstance that
it has been included in an important oceanographic work of that period by Maury,
The Physical Geography of the Sea (1st edition, New York, 1856). Croll (1870-71,
1875) refused it, but took another extreme viewpoint, since he regarded each vertical
circulation as produced by the wind. Also Carpenter (1870-77) tried to conclude
from the "Challenger" observations that a thermal circulation was present. Both agree
on the existence of a major vertical circulation and differ only on its cause.

Detailed analysis by Buchanan (1885) and Buchan (1895) of data from the
"Challenger" expedition showed that the actual spatial distribution of temperature
and salinity is incompatible with a vertical circulation of the type suggested by Lenz.
In all oceans there are alternating layers of different temperature and salinity under-
neath a relatively shallow top layer. This excludes the possibility of a single closed
circulation system with two vortices syrmnetrically placed on either side of the
equator. According to Sandstrom's proposition (p. 491) a thermo-haline circulation
is substantially promoted and intensified if the heat source is at a lower level than the
cold source, particularly when the effects of heat conductivity and turbulence are of

576

Basic Principles of the General Oceanic Circulation

minor importance as is the case in the ocean. It was mentioned that in the ocean these
heat and cold sources are at approximately the same level and that therefore conditions
are not favourable for the development of powerful circulation systems. In any case
they can be only of small vertical extent and they will be entirely incapable of filling
the whole of the oceanic space from the poles to the equator. Conditions along a
meridian will be more or less the following:

Latitude 60" 50° 40°

30° 20°

10° 0°

Predominance of heat loss due to out-
going radiation

Heat gain due to incoming radiation

Predominance of salinity decrease
(P—E > 0, melting of ice)

Salinity increase (P — E < 0, Salinity decrease through
predominance of evapora- precipitation and run-off
tion)

Since a salinity increase is equivalent to a heat loss and a salinity decrease to a
heat gain, the thermal and haline circulation will act in the same direction in the region
between the equator and in the Ross latitudes (0° until 30° N. and 30° S.). North and
south of the subtropical regions, however, they will counteract each other. A powerful
thermo-haline circulation can thus be expected only in the tropics and subtropics.
The water transport occurs towards the poles in the uppermost layer and toward the
equator underneath with an upward motion in the equatorial regions and a descending
one in the subtropics. This circulation can, however, develop only in a thin top layer
and the Lenz schematic circulation is restricted to this kind of shallow circulatory
water movement. The circulation of this tropical and subtropical top layer is dealt
with in Chapter XIX.

5. Wind Effects and the Current System in a Hydographic Circular Vortex

That the wind system of the atmosphere is also involved in the development of the
ocean circulation was not excluded by many investigators, but no agreement was
reached about the importance of its effects as long as the properties of wind drifts
were still unknown. The significance of atmospheric currents as a cause of the ocean
circulation was considerably clarified by Ekman's investigations. Probably the most
important result was to show that the wind affects directly only a top layer of not more
than 100-150 m thickness. The piling up of water at a coast by the wind will, however,
give rise to a slope in the physical sea level and to gradient currents reaching down-
wards to greater depths. In stratified water, mass compensation between upper and
lower levels (pp. 485 and 548) seems to prevent the development of deep-reaching
gradient currents. This remarkable compensation principle is readily illustrated by a
two-layered oceanic model. If in such a water mass (upper layer: pi, hi; lower layer:
p., and /72 — hi; Fig. 263) a current V is generated along AB in the upper layer, then
the physical sea level along AB will adjust itself to give a state of equilibrium between
the gradient and the Coriolis force. The deviation of the physical sea level from a level
surface ("Geoid") is denoted by Ci. Displacements of mass in the upper layer will also
disturb the equilibrium in the lower layer with a resultant mass transport in the
direction from D towards C, the internal boundary surface will decline {CD'), but

Basic Principles of the General Oceanic Circulation

577

in a direction exactly opposite to that of the sea level. This displacement of the internal
boundary surface will automatically reduce the pressure gradient imposed on the
lower layer from above. In the final equilibrium state of the lower layer there will be
no pressure gradient and therefore no motion. If io is the deviation of the internal
boundary surface from a level surface, the condition for this new state of equilibrium
is given by

^' Ci. (XVIII.4)

This simple relationship will always be present if sufficient time is available. Con-
ditions at the outer boundaries of the current aX AC and BD will be considered later
(p. 622 et seq.).

Fig. 263. Position of the physical sea surface and of the internal boundary surface of a
two-layered ocean for a forced movement of the upper layer in the interval AC-BD.

The total effect of air currents on the ocean surface can be suitably illustrated by
the simple case of an ocean uniformly covering the entire earth (no continents).
This ocean can be assumed to have two layers, an upper troposphere and a lower
stratosphere, separated by a clearly defined density transition layer. To correspond to
actual conditions in the tropics and subtropics it can be assumed further on that the
troposphere in these regions is subdivided by a transition layer at about 100 m depth
separating the top layer from the subtropospheric water masses beneath. Only zonal
(east-west) currents will be present in this hydrosphere covering the total earth and it
can be regarded as a circular vortex as described by Bjerknes (1921), centred around
the axis of the earth. The movement of the water masses in this vortex will be east-
west, and the adjacent stream lines will not influence each others. The hydrosphere
will be affected only by the atmospheric currents at the sea surface, that is, by the trade
winds between the equator and the Ross latitudes (30° N. and S.), by the west winds in
middle latitudes between 30° and 60° N. and S., and by polar east winds polarwards
60° N. and S. The oceanic movements in the individual zones of the circular vortex
and the position of the boundary surface will then be a consequence of these effects.
Since conditions are symmetrical around the rotational axis it is only required to
consider a meridional section through such a wind-generated circulation. Fig. 264

578

Basic Principles of the General Oceanic Circulation

gives a schematic representation of the water movements expected according to these
theoretical considerations.

Between 30° N. and 30° S. the north-east and south-east trade winds give rise to the
broad North Equatorial and South Equatorial Current of the Northern and Southern
Hemisphere, respectively. The maximum intensity is reached in the regions where the

Polar current

Polar, front

West winddrift

Norttiern subtropicol
convergence

North equatorial
current

Soutti equatorial
current

Souttiern subtropicol
convergence

West winddrift

Polor front

Polar current

Fig. 264. Schematic representation of the hydrosphere as a circular vortex. Current zones
and position of the main boundary surface and of the isobaric surfaces (with a strong
exaggeration of the vertical scale). {W, current towards west; E, current towards east).

trade winds are most strongly developed ; their intensity decreases toward the regions
of high atmospheric pressure in the subtropics and also towards the equator. They are
deflected 45° cum sole from the wind direction and must be associated with a water
transport towards the poles. Water will therefore be piled up at their polar boun-
daries (in about 30° latitude) and therefore a pressure gradient will be generated in
the troposphere towards the equator. Sea level and the isobaric surfaces will be de-
pressed at the equator and will rise from here towards the poles. If there is no motion

Basic Principles of the General Oceanic Circulation 579

in the water masses of the stratosphere the boundary separating it from the tropo-
sphere will slope in the opposite direction in accordance with the compensation
principle mentioned above. At this internal surface there is a stratospheric ridge at the
equator and a trough in Ross latitudes. Thus in the Atlantic the boundary is at 300 m
depth at the equator and at 700 m depth in Ross latitudes: ^2 = 400 m at 30° latitude.
With the observed values pj = 1-0260, pa = 1-0275, equation (XVIII. 1) gives the rise
in physical sea level from the equator to 30° latitude as approximately 58 cm; an order
of magnitude which agrees with the dynamic computations of the absolute topo-
graphy of isobaric surfaces. At 20° latitude where the physical sea level has a rise of
35 cm and p^— pi = 25 x 10~^, the decline of the tropospheric transition layer is
140 m, also in good agreement with observation. In this circular vortex there is no
circumstance which would give rise to an equatorial counter current.

Winds in the atmospheric West Wind Drift are of rather variable character; but
only in the general average westerly winds predominate. In the top layer they produce
an oceanic West Wind Drift and a consequent piling up of water cum sole towards
the subtropics, which counteracts the accumulation of water associated with the equa-
torial currents. There is thus an accumulation of water from both sides in a belt around
the earth. On the equatorial side of this belt water flows westward, on the polar side
eastward. This is the subtropical convergence region, one of the most important bound-
ary lines of the oceanic circulation. Corresponding to the downward slope of the
physical sea level towards the poles there is an upward slope in the internal boundary
surface between the troposphere and the stratosphere from its deepest position in the
subtropics to the surface of the ocean at the polar front {polar convergence). This is the
60° N. and S. it must rise 700 m over 30 degrees of latitude. When pi = 1.0265 and
P2 = 1.0275 the physical sea level will have a slope of 68 cm according to equation
(XVIII. 1). If the physical sea level at the equator is taken as zero, it will have an eleva-
tion of 58 cm in the subtropics and a depression of 10 cm at the polar front. The
prevailing easterly winds around the polar caps produce a westward drift current
(polar currents) and there is a corresponding rise in the sea level from its lowest
position at the polar front.

Although the circulation system shown in Fig. 262 is only schematic, it shows the
main features of the surface circulation system clearly, particulary as in the Pacific
and in the circumpolar Antarctic waters where it is not strongly disturbed by the
presence of continents. With a circular vortex of this type under stationary conditions
no vertical movements are to be expected. A deep-sea circulation will therefore not
develop and the three horizontal current zones (the Equatorial Currents, the West
Wind Drifts and the Polar Currents) can be explained as solely caused by winds.
The topography of the physical sea level, of the internal boundary surface between
the troposphere and the stratosphere and of the tropospheric transition layer of the
tropics and subtropics are coupled with these zones.

6. The Influence of Meridionally Oriented Coasts on the Oceanic Circulation

The oceans are bounded everywhere on their western and eastern sides by conti-
nents which act as meridional barriers to the oceanic circulation and prevent the
formation of a simple circular vortex around the earth. At the meridional barriers
the equation of continuity must be satisfied, and in order to allow the conservation of

580

Basic Principles of the General Oceanic Circulation

mass, meridional currents must develop that will determine the nature of the circula-
tion. It appears that these boundary conditions are more easily fulfilled for a sea with
a meridionally oriented eastern coast than for one with a meridionally oriented
western coast.

{a) Conditions West of a Meridionally Oriented Coast

SvERDRUP (1947) has shown that a steady state solution can be found for a density-
layered ocean by starting at a meridional boundary and working westwards even when
frictional effects are neglected. In the vorticity equation (XVII. 5) the wind stress vort-
icity must be balanced by the planetary vorticity alone and, as shown already in XVII.3
second of the major boundaries of the oceanic circulation. To reach the surface at
the boundary conditions and the equation ofcontinuity(XVII.4) determine the currents
westward from the meridional boundary (east coast). For a purely zonal wind
{Ty = 0), the mass transports (omitting the first term of (XV1I.7) ; lower latitudes)
will be given by

My = --^^' and M^^j-^. (XVIII.5)

Assuming in a schematic way according to actual conditions in the ocean (equator
to 30°: easterly winds; 30° to 60°: westerly winds)

T =

a sm -r-y.

(XVIII.6)

where / is the distance from the equator until 60°, then

-^.^-jT^^njy.

From this it is easy to derive the following table of signs of the different quantities for
an eastern or western meridional barrier.

Barrier to the east

Barrier to the west

0-1/

il-il

1/ - 1/

0-il

y - ii

^/-f/

—

- c

» +

d^TJdv^

—

— 1

—

M. .

• ! ~

—

+ c

► +

Possible case

Impossible case

West of the barrier, T^ and M^ are, according to (XVIII.5), both positive or both nega-
tive. However, east of the barrier they are of opposite signs, which is impossible.
The equations (XVIII.5 and 6) give a steady state solution only for a sea area to the
west of the boundary. The foUov/ing example can be taken as an illustration of such a
solution.
Selecting T^ = — 0-4 sin 6(f> dyn cm"', gives

2-4
2<x) cos (f>

cos 6(f) ; Mx

14-4 Zljc
2Rw cos <f>

sin 6(f) and tfj = —

2-4 Ax

2co COS(f>

cos 6(f).

Basic Principles of the Geiieral Oceanic Circulation

581

Fig. 265. Stream lines of the flow representing the field of mass transport; differences of the
values of the stream function between two stream lines represent the net mass transport
in 10® metric tons per second flowing between these stream lines (from the surface down to a

depth of no motion).

Figure 265 shows stream lines of flow representing the field of mass transport. The
principal troughs and ridges are accounted for by the wind stress function. Off the
coast in the east the currents are weak and the meridional component is directed south-
wards in middle latitudes.

The integrated equations give no information on the distribution of vertical motions
in the deep oceanic layers. A better comprehension of these currents can be gained by
accurate calculations for the very simple model of Sverdrup. Stommel (1957) has
recently given a very instructive description of such a case, in which zonal wind stress
was assumed to act on a homogeneous ocean surface with an eastern coast line.
Figure 266 shows the solution. At the surface there is a zonal wind stress with a similar
distribution as that shown in Fig. 265. The stream lines will therefore also be similar to
those in the diagram. The transport in the thin Ekman layer, indicated by the upper
arrows, will produce a vertical downward velocity in the central part of the diagram.
Outside the zonal belt of westerly winds the vertical velocity will be directed upwards.
These vertical components from the bottom of the Ekman layer to the bottom of the
ocean decrease linearly to zero. The divergence and convergence system of the meri-
dional components of geostrophic velocity are coupled with this vertical velocity field.
At the latitude of maximum westerly wind, where there is no impressed vertical
velocity, the geostrophic flow will be entirely zonal and will decrease linearly towards
the eastern coast. The topography of the physical sea surface, which determines the
pressure field associated with the geostrophic flow, is also shown in Fig. 265.

582

Basic Principles of the General Oceanic Circulation

If in addition bottom friction of the type described by Ekman is taken into account,
the current field will be slightly altered; now the bottom current must aiso contribute
in order to satisfy the convergences and divergences appearing in the current field of
the Ekman top layer.

Fig. 266. Sverdrup-type solution in a homogeneous ocean of uniform depth, bounded by a
meridional coastal wall on its eastern side. The wind system with sinusodial pattern is
indicated by shaded arrows hovering above the surface. The curved lines with arrows
are isobars and give the direction of the geostrophic horizontal flow (independent of the
depth). At a number of subsurface depths the velocity components are shown by solid
arrows (according to Stommel 1957).

Considerably more complicated models of this type can, of course, be developed,
but they will all show that the boundary conditions at any coast to the west cannot be
satisfied except by taking into account processes involving the dissipation of energy.

(b) Conditions East of a Meridionally Oriented Coast

In the western part of the oceans, and particularly along the western boundary, the
vorticity related to lateral friction must also be taken into account with an additional
term in order to satisfy mass conservation and space continuity conditions in the
vorticity equation (XVII. 5). With this equation Stommel (1949, 1951) was the first to
give an explanation of the westward intensification of ocean currents. He took the case
of a symmetrical anticyclonic wind circulation over a closed rectangular oceanic area
in the Northern Hemisphere. The wind stress vorticity is thus negative over the entire
ocean. The effect of the wind stress can be expected to cause an anticyclonic circula-
tion in the sea. The horizontal eddy viscosity will tend to counteract the effect of wind
stress. In the western parts of this ocean the anticyclonic flow will transport water
northward, in the eastern parts southward; in equation (XVII. 5) the planetary vorticity
effect is therefore negative at the western side of the ocean and positive at the eastern
side. This is a consequence of the conservation of angular momentum or, what amounts
to the same, of the variation of Coriolis parameter with latitude.

Basic Principles of the General Oceanic Circulation

583

If the absolute numerical values of the three vorticity terms in (XVII. 5) are denoted
by a, b and c, then for a symmetrical wind system, {a) would be negative and would
have the same numerical value in both east and west. For an equal velocity, a sym-
metrical oceanic circulation would require an equally great frictional vorticity;
(Jb) would thus be positive and have the same numerical value in the east as in the
west. The planetary vorticities in the west and in the east would also have the same
numerical value but are of opposite signs. Thus

Off the western boundary
— « + Z7 - c =0

Off the eastern boundary
— a + Z) + c =0

These requirements are satisfied only when c = 0, that is, when there is no meridional
transport, and are therefore incompatible with the conservation of mass. This is a
qualitative explanation

(1) of the impossibility of a symmetrical circulation in association with a sym-
metrical wind field,

(2) of the impossibility, mentioned above, of deriving a suitable circulation off the
western coast of an ocean without accounting for frictional influences.

As shown by Stommel, an anticyclonic circulation is possible in the case just
discussed only when the water transport off the western boundary is substantially
intensified and the lateral shearing stresses consequently, of course, increased corres-
pondingly. To illustrate this, Stommel gives some arbitrary values for the vorticity
terms in an asymmetric circulation. These are shown in the following Table 149.

Table 149. Vorticity tendencies in an asymmetric
circulation

Strong northward Southward flowing
flowing currents current over the
in the western edge rest of the ocean

Wind stress (a) .
Frictional (b)
Planetary (c)

- 10
+ 100

- 90

-10
+01
+ 0-9

Total

Among the interesting consequences of this theory are :

(1) the fact that although energy is added to the oceans by work done by the wind
over the entire surface, it is dissipated primarily in the strong western currents ;

(2) that a good representation of the circulation in the zonal currents of westward
or eastward direction can be obtained independently of friction from a know-
ledge of the wind stress field alone.

MuNK (1950) was able to evolve a comprehensive theory of a wind-driven ocean
circulation by combining three new concepts :
(fl) the introduction of lateral stresses associated with the horizontal exchange in

large eddies (Defant, 1926; Rossby, 1936a),
(6) the possibility of computing currents in baroclinic oceans from the known

wind stresses (Sverdrup, 1947), and

584 Basic Principles of the General Oceanic Circulation

(Stommel, 1948) which makes it possible to explain the westward intensification

of a wind-generated ocean circulation.

This theory accounts for many of the major features and some of the details of the

general ocean circulation on the basis of known mean annual winds. Briefly the

fundaments of this new theory are:

The vorticity equation (XVII. 5) can be put into a practical form by the introduction
of expressions for the lateral frictional forces. According to (XL 13 and 14) these
frictional forces have the form

(d^u dhi\ , IdH cH\

^- = '^ (a? + 8/) ^°^ "' = ^ (a? + if) ■ (^^"")

A is the lateral eddy viscosity pertaining to horizontal shear v*'hich is presumed to be
constant and horizontally isotropic, neglecting variations due to differences between
zonal and meridional motion of large horizontal vortices on a rotating earth. Intro-
duction of these expressions into (XVII. 5) with the stream function according to
(XVI. 25), gives the differential equation for mass transport

AV^ - iS ^\^ = - curL T, (XVIII.8)

where V^ is the biharmonic operator (see XVI.26) and curl, Tis the vertical vorticity
component of the wind stress. It can be shown, in accordance with the relationship

lateral stress curl + planetary vorticity +

western solution + wind stress curl = 0

(XVIII.9)

central solution J

that in the central and eastern oceanic areas the planetary vorticity and the wind-
stress curl have opposite signs, resulting in balance in which the lateral stress plays a
negligible part. Along the western boundary the planetary and the wind-stress curl
have the same sign, and the lateral-stress curl balances both, planetary vorticity and
wind-stress curl. It can be verified that in this region the wind-stress curl is numerically
unimportant although it is, of course, the primary cause of the circulation.
To equation (XVIII. 7) must be added the boundary conditions

^- = 0; (yj-O, (XVIII.IO)

boundary \ / boundary

where v is normal to the boundary. The first equation states that the boundary itself
is a stream line, the second that no slippage occurs against the boundary.

Munk assumed :

(1) a rectangular ocean extending from x = 0 to .v = r and from y = —s to y = -\-s.
The boundary conditions will then be

0 = dijjjdx = 0 for ;c = 0 and x — r "\ rxVTlT 1 H

0 = dxltjdy — 0 for_y = —s and y = A^s j

Basic Principles of the General Oceanic Circulation 585

(2) a zonal wind circulation (T y= 0); for this the stress on the ocean surface in the
interval —s < y < -hs can be given as a Fourier series, a general term of which is

T^^^ = c + aco^ny + b sin ny with n = j^-; (j =1,2,...) (XVIII.12)

The solution of (XVIII. 8) which satisfies the boundary conditions is 0 = — rXfS-'^ curl, T
whereby

/ 1 \ r 2 ikx /

2 -*A^---/V3_ _. ^ J

1
kr

kx — e-''(^-^)

west ^ , ' j.(XVIII.13)

central

^ , '

east

Here k is the "Coriolis friction" wave-number which has the vale ^(fijA) and is
assumed to be constant. The solution is valid as a first approximation when
y = (njky <^ 1 and g-''" < 1. When ^ = 0-016 km-^ and r = 6000 km the value of
the stream function ip will be accurate within 10%, if y < 0-25, corresponding to a
minimum zonal wavelength, lir/n, of about 1500 km. Since for the mean annual
stress distribution the shortest wave length of the important north-south variations,
the distance between the northern and southern trade winds amount to 4000 km, the
approximation leading up to (XVIII. 13) therefore appears to be valid for a study of the
general ocean circulation in relationship to the general atmospheric circulation.

A knowledge of the wind distribution over an ocean thus permits a direct quanti-
tative calculation of the current field in the ocean. It was calculated by Munk for the
North Pacific, first as an approximation for a rectangular ocean, and later for a tri-
angular ocean (Munk and Carrier, 1950), which gives a better representation of
actual conditions.

The solution (XVIII. 13) shows in the first place that the zonal wind system divides
the ocean circulation into a number of gyres. The dividing lines between them lie in
the latitude of maximum west wind, in the northerly and southerly trade winds and in
the doldrums. The latitudinal axis of each gyre may be defined by d^TJdy"^ = 0. The
Atlantic Sargasso Sea is associated with the inflection point in the mean wind stress
curve between the westerly winds and the north-easterly trades. The inflection points
between the doldrums and the northern and southern trades determine the boundary
of the equatorial counter current.

When Xis computed from (XVIII. 13), it is found that the equations fall naturally
into three parts, each of which dominates in a given sector. At the western edge of the
ocean x <^ r, and becomes

Xwest = \ e-^- cos (^ ^^ - ^) + 1 (XVin.l4)

representing slightly "underdamped" oscillations with a wavelength given by

586

Basic Principles of the General Oceanic Circulation

A remarkable feature is a counter current east of the main current, with a magnitude
of 17% of that of the main one. There can be little doubt that such counter currents
exist, although this fact has been obscured in some instances by the smoothing of
data. This theoretical result has in fact been shown to be in agreement with observa-
tions (see p. 536 et seq.).

The total transport of the western current and counter current is found by putting
numerical values of A' into X.14) giving

"Av

M7/-;8-icurl, r.

(XVIII. 16)

The resulting expressions are independent of A and the transport can be computed with
a relatively high degree of accuracy; the uncertainty is of the same order as that in the
calculation of wind stress. Table 1 50 gives a comparison between the transport values
of some western currents determined from oceanographic observations, and those
computed from the zonal wind stress using equation (XVIII. 14). The two sets of values
are of the same order of magnitude, but the calculated transport values differ from the
observed values by a factor of as much as two ; the discrepancy is not surprising when
it is considered that amongst other uncertainties the wind and current data are not for
the same year, nor necessarily for the same time of the year. Another source of error
may be due to possible underestimation of the wind stresses at low wind speeds. It
can be assumed, in accordance with views held at the present time, that the dependence
of wind stress on the wind velocity is given by /c = 0-0026 at high wind speeds and
K r-^ 0-008 at low speeds, with the discontinuity at Beaufort 4 (see p. 421 and especially
MuNK, 1947). This assumption, however, does not appear to be absolutely certain
and further investigations are required.

Table 150. The mass transport of some western currents determined from the
wind stress and from oceanographic observations

Current

Lat.

1013^

(cm-1 sec~^)

(km)

101° {8T,ldy)
(g cm-2)

101-^

by wind stress

(g sec-i)

Ocean, obs.
(g sec-i)

Gulf Stream .
Kuroshio
Oyashio C.
Brazil C.

35° N.
35° N.
50° N.
20° S.

1-9
1-9
1-5

2-2

6500

10000

5500

-15

-20

-6-5

-5-8

74* (55)t
65*

-1%

-5 to- 10*

* Sverdrup et al. (1942), pp. 605, 761.

t Adjusted for a supposed southward motion of 19 x 10^^ g of slopewater.

+ For August (Uda, 1938).

Away from both boundaries the stream-line function X reduces to

-^central = 1 "

which gives the central oceanic drift; this is a broad constant drift that compensates
for the swift shallow western currents. Equation (XVIII. 17) also gives

(XVIII. 17)

(XVIII. 18)
which agrees with the relationship derived by Sverdrup (see p. 580, equation XVIII. 5).

Basic Principles of the General Oceanic Circulation
In the eastern part of the ocean, the eastern solution is valid in the form

X J_
r kr

\ — g-k(r-x)

587

(XVIII. 19)

It represents an exponential slippage zone with a width of approximately rr/k.

If A

10' cm^ sec-\ the width will be about 200 km.

The complete circulation of an ocean shows pronounced east-west asymmetry.
The westward intensification of ocean currents is an effect of the planetary vorticity.
The asymmetry may be expressed by either of the ratios :

My (west, cur.)

—0'55kr or

(XVin.20)

My (cent, cur.) "' x (west. cur. axis)

that is, by either the ratio of the maximum western current to the central drift, or by
the ratio of the width of the ocean to that of the western current. The asymmetry
increases with r, decreases with A and </>; for the Atlantic kr ^ 100.

Along the western coasts of the continents there are relatively strong seasonal ocean
currents (California Current, Benguela Current, Peru Current), which cannot be
explained by the simple assumption of zonal winds. To cover these currents which are
also essentially dependent on winds, the theory must be expanded by the introduction
of corresponding meridional wind stresses. This solution also has been given by
Munk together with a general solution in which is introduced a general field of wind
stress associated with the large-scale atmospheric circulation.

To demonstrate the ability of this theory of the general ocean circulation to express
the actual mean current conditions in an ocean, a theoretical solution for the Pacific
as an approximation for a triangular ocean is given for comparison with a recent
representation of currents based on observations in Figs. 267 and 268 (cf. Munk and
Carrier, 1950).

It can be clearly seen that all the essential features of the current patterns are covered
by the theory.

There is no doubt that the Stommel-Munk theory of ocean circulation explains the
large-scale geographic picture of the horizontal ocean currents in all oceans as a direct
effect of the permanent wind system over these oceans. There is very good qualitative
agreement between the water transport computed from wind distribution and that

Fig. 267. The computed mass transport in an ocean of triangular form represented by

stream lines. Between two neighbouring stream lines 6 million tons of water flow in the

direction of the arrows per second.

588

Basic Principles of the General Oceanic Circulation

Fig. 268. The oceanic mass transport of the North Pacific Ocean, derived from data
available. Between two neighbouring stream lines 6 million tons of water flow per second.
(1) Kuroshio; (2) Oyashio; (3) Alaska Current; (4) California Current; (5) Sub-Antarctic
Current; (6) North Pacific Current; (7) East Pacific Vortex; (8) North Equatorial Current.

deduced from oceanographic observations, and this agreement is confirmed by all
investigations that have been carried out along the lines of Munk's computations.
HiDAKA (1950, a, b, c, 1951) has dealt in particular with the wind-generated ocean
circulation of the Pacific and has obtained an overall climatological oceanic circula-
tion, that fits admirably with that deduced from ship's displacements. His mathe-
matical treatment of the problem differs from that used by Munk only in taking
higher order terms into consideration and in using infinite series for the solution of
the differential equation, in some instances with spherical co-ordinates, while Munk
and his collaborators have used planar co-ordinates. More recently, Hidaka (1955)
has presented a detailed numerical theory of the general circulation of the Pacific
which he regards as a purely wind-generated phenomenon. He uses the assumption
that the vertical velocity vanishes exactly at all points. Further, he gives the horizontal
distribution of the stream lines for different subsurface levels. These circulation
patterns are all similar to the sea surface circulation. The only noticeable difference is a
general reduction in intensity of the movement with depth. It may be already as little
as half the surface intensity in 250 m depth. His numerical results are, however,
difficult to interpret on a physical basis, and appear insufficient for an explanation of
the vertical mass transports necessary for continuity.

Hansen (1951, 1954) treated the circulation problem as a boundary value problem
("Eigen" value problem). His method is equally suitable for finding the volume trans-
port and the form of the sea surface in an enclosed part of the ocean from the known
wind field. Hansen calculated the volume transport and the sea surface topography for
the equatorial part of the Atlantic from the average August wind field, and obtained
a satisfactory agreement with results based on observations of ship's displacements
and of the density distribution.

While for all methods the agreement is very good qualitatively, this is not always so
quantitively. Munk, for instance, obtained transport values for the Atlantic and the
Pacific which were only half as great as those computed from observational data
(36 and 39 x 10^ m^/sec for maximum transport by the Gulf Stream and the Kuroshio,
respectively, against observed mean values of 55 to 74 and 65 x 10^ m^/sec, res-
pectively). It is not improbable that the discrepancy arises from the fundaments of the
theory, possibly from the use of the mean wind stress based on climatological wind

Basic Principles of the General Oceanic Circulation 589

charts without taking into account the deviations. It might also be due to the imper-
fections in the present knowledge of the relationships between wind velocity and wind
stress (see pp. 421 and 586) or due to the use of plane co-ordinates instead of spherical
ones for the calculation of conditions on the curved surface of the earth. It is note-
worthy that Hidaka has obtained good numerical agreement for transport in the Kuro-
shio Current using spherical co-ordinates. The most probable reason however is that
the actual dynamics of the strong western boundary currents (such as the Gulf Stream
and the Kuroshio) are left essentially unexplained by the Stommel-Munk theory. In
order to explain the narrowness of these boundary currents it is necessary to take an
eddy viscosity so large that the eddy sizes would be comparable to the width of the cur-
rent. This can never be the case. Pressure inertia and the variations of Coriolis para-
meter with latitude all seem to play an important part in the dynamics of these boundary
currents (see p. 550). It is striking that there is no indication of a "westward intensifi-
cation" of ocean currents in the Southern Hemisphere; the Brazil Current and the
East Australian Current for instance are not so strongly developed along the east
coast of the continents as the Gulf Stream and the Kuroshio. It wouid be expected
that if the planetary vorticity were the only cause of the westward intensification in the
oceans of the Northern Hemisphere it would show the same effect in the South Atlantic
and South Pacific. It appears however that the vertical structure of the ocean also
plays a role in the theory since the depth d is correlated with the oceanic structure and
the magnitude of d cannot be chosen arbitrarily, d denotes the depth over which an
integration has to be performed in order to eliminate the effect of the vertical oceanic
stratification and of internal vertical friction. Usually the depth of no motion has been
taken as d and only the horizontal velocity of the water movement has been taken in-
to consideration; the vertical velocity is presumed to be zero or so small that it can
be neglected. This assumption is certainly incorrect and may lead to an entirely false
picture of the horizontal circulation. Stommel (1956) has given a detailed discussion
showing that the existence of a level of no motion in the ocean where all the three
velocity components vanish cannot be substantiated; in fact the maximum vertical
velocity occurs at the depth of no meridional \Q\ocity (see p. 499). A paper by Neumann
(1955) is of interest here. He has re-examined the theory for a horizontal wind-driven
ocean current taking into account the spherical shape of the earth the average vertical
density stratification and the variable depth of the lower boundary of the circulation
system. The latter assumption is the same as the assumption that the depth d is the
depth of the layer of no meridional motion. Integration of the usual equations of
motion for the geostrophic wind taken over the depth _ between +^ and —d and with
P = p(x,y,z) gives the equations of transport

dP 81, cd,

■^ -^ cy ^^^^ cy ^^ ^ dy

CP CL dx.

(XVIII.21)

Introducing

' T+d

p(-) dz; P(-d) = gp(i + rf) and P = p dz = 2f (? + df

590 Basic Principles of the General Oceanic Circulation

and taking into account that the divergence of the total mass transport is zero and
C <^ d, one obtains

/dxdp dddp\ Idxdl dddr\ ^ ^ ^

and from the second equation

fM, = Igd' ^ + gpd^^ (XVIIL23)

Equating M^ in (XVIII.22) and (XVIII.23) gives

/^ 1 8d\ 8C/Pd 8d\ I8p nSC\ 8p\ 8d _

\f~ ddy) dx-^ [jl" d^rpdx^ [ddy-^ -p dyjdd " ^- (^^111.24)

In the case of a homogeneous ocean (p = const.), equation (XV1II.24) reduces to

B 1 8d\8^ 1 8C 8d

This equation states that in the case of a constant depth d only zonal steady currents
are possible, because the first term will vanish only when 8l,j8x = 0. When the
depth J is variable, all current directions are possible, if d satisfies certain conditions
according to (XVIII.24). If the depth d is a function only of y (the latitude), then,
provided that 8d/8y ^ 0

B 1 8d

This equation is identical with (XVI. 19) and states that for stationary currents the decline
of the lower depth d of the current system towards the poles must follow a law
d — K?.m 4>. In a stratified ocean (p = p{x,y,zy) the interrelationship is more compli-
cated. Equation (XVIII.24) shows, however, that for a constant depth t/ of no horizontal
motion, there can be no meridional mass transport due to frictionless currents, since
when d = const., the equation reduces to

(XVIII.27)

On substitution in equation (XVIII.23) it is found that M^ = 0.

It has been shown above (p. 497) that in the Atlantic Ocean the zonal mean of the
depth of no meridional motion follows the above equation. This can be interpreted
to mean that the planetary vorticity {^My) is compensated by a corresponding balanced
topography of the lower boundary of the current system. This is frequently the case
in the South Atlantic, and here the westward intensification, which of course is a
consequence of the planetary vorticity effect, is only weakly developed.

In criticism of Neumann's arguments, Stommel has questioned the assumption
that the depth f/ is a depth of no motion, and has pointed out that on the contrary, the
greatest vertical velocities occur at this depth. Neumann's equations can also be
derived from the basic assumption that the potential vorticity in the large-scale

Basic Principles of the General Oceanic Circulation 591

oceanic circulation is constant (see p. 336); that is, dldt{t, -'rf)/dc, — 0, where ^ is
here the relative vorticity. Since generally ^ < /, this equation reduces to

dy \ d I

for stationary predominantly zonal currents. From this it follows, since C <^ /, that

/ \8d I 8f p

^~ const, or -^ =--/=-„ (XVIII.28)

d ddy fdy f

which is equation (XVI1I,26). However, the assumption of constant potential vorticity
is valid only for horizontal geostrophic currents, but does not hold when vertical
velocity components are also present. Stommel's objection seems then to be justified
and Neumann's equations are valid as a first approximation only when the vertical
velocities are small compared with the horizontal ones.

Chapter XIX

The Tropospheric Circulation

1. The Position and Structure of the Oceanic Troposphere

The important subdivision of the oceanic space into troposphere and stratosphere is
due primarily to the climatic influence of the atmosphere on the water masses of the
uppermost ocean layers. More or less constant conditions in weather and radiation
at the ocean surface give rise to the development and maintenance of water types of
diff'erent character in different climatic zones. Broadly speaking there are two principal
water types which are constantly being formed in large quantity and with a rather
constant internal structure; they correspond to the two great zones of contrasting
climate, the tropical and subtropical regions, and the polar regions. These two water
types are:

(1) the tropical-subtropical water type which is warm due to the excess of incoming
radiation and has a high salinity due to evaporation, and

(2) the cold weakly saline water type of the subpolar and polar zones.

The former is lighter, the latter heavier, and this difference is the cause of con-
tinuous large-scale movements. These movements follows the fundamental principle
that each water type tends to flow by the shortest route, by vertical or horizontal dis-
placement to the depth in the ocean at which it will be in a stable equilibrium corres-
ponding to its density; here it will spread out as a layer. The heavier subpolar water
type therefore sinks to greater depths, and spreads more or less horizontally to fill in
this way the deep lower layers of all the oceans. The lighter tropical and subtropical
waier type, on the other hand, remains in the upper layers of its original zone as the
lightest water type. The subdivision in the structure of the oceans is thus a con-
sequence of circulation. It is to be expected already from the history of formation of
the two main oceanic subspaces, that they will have essentially separate circulations;
these will be called tropospheric and stratospheric. This does not imply that there is
no connection between the two circulations; on the contrary, at certain places inter-
actions occur and the water masses of both type undergo transformation by turbulent
mixing and manifold atmospheric influences so that tropospheric water becomes
stratospheric and vice versa.

The thermo-haline structure of the troposphere has been explained in pt. I, Chapter
III, §4, p. Ml et seq. and IV §3, p. 165 et seq. The most important phenomenon is the
layer of discontinuity in the vertical distribution of temperature and density which is
always sharply defined in the tropics and subtropics and is associated with a charac-
teristic salinity distribution. An example is shown in Fig. 70 of pt. I. Beneath the dis-
continuity layer which acts as a barrier to upward and downward movement, is the
subtroposphere which is occupied by little differentiated and nearly motionless waters.

592

The Tropospheric Circulation

593

It is usually difficult to fix a definite boundary between the troposphere and the
stratosphere. In the vertical density profile it appears as a slight intensification of the
vertical gradients; but often it is quite indistinct because of the very great distance
between observation levels at these depths. It should probably be referred to only as a
boundary layer. An approximate boundary can be obtained using the oxygen content
as a criterion (Wust, 1936Z?, see pt. I, p. 66 et seq.); it is then defined by the inter-
mediate oxygen minima. The method is based on the assumption that these minima
indicate layers where the air supply is least, that is, those localities where the renewal
of the water masses is particularly slow and where horizontal movement of the water
is entirely missing. It has frequently been pointed out (p. 494) that in the uppermost
layers the position of the oxygen minima is affected by biological processes. However,
oxygen minima can be used at greater depths to specify approximatively the different
circulations. In the Atlantic the oxygen minimum extends across the 1 10 degrees of
latitude (from 45° S. to 55° N.) between the oceanic polar fronts of both hemispheres;
its mean depth along a meridional section is given in Table 151.

From the Southern Hemisphere polar front the lower limit of the oceanic tropo-
sphere sinks rapidly down to 600 m in the southern convergence region (between 35°
and 25° S.), and rises again to about 300 m in the tropics. Just north of the equator, it
is at first somewhat irregular and then sinks gradually down to about 950 m in the
northern convergence region (30° to 40° N.).

Reasonably accurate data are available for the tropospheric circulation which
extends throughout the space between the sea surface and the lower boundary of the
troposphere. Defant's (1936c) representation of conditions in the Atlantic also
includes subsurface data over the whole area. For the other oceans the series observa-
tions are sufficient for interpretation only along single meridional or zonal sections.
No major differences between the oceans in the principal features of circulation are to
be expected.

Table 151. Lower limit of the troposphere in the Atlantic Ocean
(Determined from the position of the oxygen minimum. Depth in metres.)

Section

50°

45°

40°

35°

30°

25°

20°

15°

10°

5° Equa-
tor

(1000)

850

830

820

770

550

280*

350

Western section

400

1 _
■i

—

400

500

550

600t

580

450

300

280*

450

790

IIQ

830

880t

870

680

470

380 i 330*

Central section

400

—

(100)

320

500

600t

580

550

420

300*

400

(900)

(900)t

(250)

820

680

(550)

520

400

Eastern section

350*

—

300

470

530t

510

450

380

300*

390

400

N. Northern Hemisphere; S. Southern Hemisphere

* Minimum values; f Maximum values.
Q

594

The Tropospheric Circulation

2. The Tropospheric Circulation of the Tropical and Subtropical Oceans

The tropical and subtropical circulation of the oceanic troposphere is dominated
by the enormous water transports of the North and South Equatorial Currents.
They determine dynamically the position of the tropical and subtropical discon-
tinuity layer. Its depth in the Atlantic between 25° N. and 25° S. is shown in Fig. 269.
From a depth of more than 200-300 m in western Ross-latitudes of both hemispheres

Fig,

269. Depth (m) of the tropospheric discontinuity (thermocline) in the Atlantic Ocean
between 25° N. and 25° S.

the discontinuity layer rises towards the southeast to a depth of 40 m in the Northern
Hemisphere and towards the north-east to a depth of 20 m in the Southern Hemi-
sphere. Between the equator and about 6°-10° N. these rising slopes are separated by
an east-west depression extending into the Gulf of Guinea. This striking arrangement
of the topography of the discontinuity surface is a direct consequence of the equatorial
currents on either side of the equator; because of dynamic reasoning these currents
also determine the rise of the discontinuity layer towards the equator. Up to about
6° to 10° N., the depth of the density transition layer is associated with the Equatorial
Counter Current and its further extension (the Guinea Current). For a connection
between the state of motion of the water masses above and below the discontinuity
and the topography of the discontinuity layer see p. 463 et seq. Further information on
the conditions of motion in the individual layers of the oceanic troposphere can be
gained by investigation of the striking salinity maxima near the discontinuity layer,

The Tropospheric Circulation

595

that intervenes between the homo-haline and weakly saline top layer and the deeper
lower salinity layers with an equally low salinity. Study of the position of these maxima
and their development showed that they intrude under the less saline top layer from
the extensive subtropical accumulations of highly saline water to the north and south.
These intrusions spread along preferred paths, the location of which throws some light
on movements within the middle and lower layers of the troposphere. This spreading
and its dynamics have been discussed in pt. I, Chapter IV, p. 166. There Fig. 72
(p. 168) shows that the salinity maxima are present everywhere except in two narrow
bands in both hemispheres where the density transition layer comes closest to the sea
surface. Evidently, the horizontal extension of the highly saline intermediate layer is
cut short in this region, and here the water masses must be deflected upward. The
region between the two bands without salinity maxima lies in the Equatorial Counter
Current. Here the supply of water that forms the salinity maxima comes from the
west, from regions which are not reached by the bands free from the salinity maxima
and are fed here from north and south. From these facts it is possible to derive a
three-dimensional system of currents in the oceanic troposphere of the tropics and
subtropics, that is illustrated schematically by the meridional section in Fig. 270.

20° S 15

20° N 25°

200^

Fig. 270. Schematic representation of the zonal and meridional velocity components of the

tropospheric circulation in the Atlantic Ocean (the topography of the thermocline is

exaggerated in the vertical scale by about 1 :1 million; that of the physical sea surface even

more); W, current towards west; E, current towards east.

Where the stream lines are divergent in the top layer they are convergent in the dis-
continuity layer; the two bands with a low salinity are thus regions of upwelling water.

The zonal components of motion do not appear in the meridional section and it
should not be forgotten that these are considerably more important. Compared with
these the transverse circulation is rather weak. This transverse circulation is primarily
a thermo-haline circulation and is the consequence of the internal forces of the mass
distribution (p. 575). It involves only the top layer down to the density transition layer
and in the strong zonal motions of the wind-driven equatorial currents it can hardly
be detected. It is, however, responsible for the pronounced vertical and horizontal
salinity distribution that is characteristic for the uppermost layers of the tropical
oceans.

The water masses beneath the density transition layer (in the subtroposphere) are
very uniform and colourless and the water movements here must therefore be very
weak. Since they lie beneath the barrier, they can be only slightly aff'ected by turbulence

596

The Tropospheric Circulation

and convection and they have an extremely low concentration of oxygen which is
largely due to the almost total stagnation and also due to biological causes.

The internal forces, providing the motive force for the entire current system of the
tropics and the subtropics, are produced, on the one hand, by the wind system present
in these zones and on the other hand, by the internal pressure field set up by the thermo-
dynamic conditions. Figure 271 shows the absolute topography of the physical sea level
in the Atlantic Ocean pictured by isobaths drawn at intervals of 5 dyn/cm between
35° N. and 35° S. and at intervals of 10 dyn/cm outside this area (Defant, 1941^).
The direction of this stationary gradient current, which corresponds to this pressure
field is indicated by arrow-heads on the dynamic isobaths. Comparison of this topo-
graphy in the tropical and subtropical area with that of the tropospheric density transi-
tion layer (Fig. 269) shows that they are almost mirror images; in deeper layers the

Fig. 271. Absolute topography of the physical sea surface (dynamic isobaths drawn from
5 to 5 dyn cm, 10 to 10 respectively).

The Tropospheric Circulation

597

pressure surfaces are of the same form as the sea surface but the pressure gradient
decreases rapidly with depth (Fig. 272). The lower limit of the tropical and subtropical
circulation must lie at the 500 decibar surface where the pressure gradient is almost
zero; already at 200-300 m depth the velocity of the currents is very slight and the
Equatorial Counter current does not reach nearly as deep as this (approx. down to
1 50 m). A comparison of the topography of the physical sea level and the gradient

ra° W

Fig. 272. Absolute topography of the 100-decibar (upper picture) and 500-decibar surface

(lower picture) of the subtropical and tropical region of the Atlantic Ocean (dynamic isobaths

are drawn from 2-5 to 2-5 dyn cm).

598 The Tropospheric Circulation

currents at the sea surface derived from it (see ''Meteor''' Report VI §2, supplement 22)
with current charts derived from observations shows that the trade winds are the main
cause of the currents in the uppermost layer of the sea. These give rise to a total water
transport at right angles cum sole of the wind direction. In the Northern Hemisphere
the water '^flows towards west-north-west and in the Southern Hemisphere towards
west-south-west. Along the east coasts of continents and also at the eastern boundary
of the strong water displacements, which are directed from north to south along
the coast lines, water is accumulated and piled up and thus a pressure gradient is
created to the south-east in the Northern Hemisphere and to the north-east in the
Southern Hemisphere. This is shown clearly by the topographies of the pressure sur-
faces and of the sea surface, respectively. In the trade-vv-ind region the resultant ocean
current is then no longer solely due to the effect of the permanent air currents charac-
teristic for these latitudes, but is also affected decisively by the mass distributions in
the uppermost layers. A diagram of forces for the central part of the South Equatorial
Current according to the "Meteor" observations, has already been discussed (Fig. 180,
p. 424). It allows an estimate to be made of the effect of the individual forces in the
formation of this major current. It is of particular interest that the water masses in the
equatorial currents^ow against the slope of the physical sea level and the pressure surfaces,
that is to say, uphill. Part of the force transferred to the water by the winds is used
in overcoming this gradient, so that the velocities of the water displacement are
correspondingly somewhat reduced.

The pressure field associated with the Equatorial Counter Current is clearly shown
in the topography of the physical sea level (Fig. 271) and in the topography of the
isobaric surfaces (Fig. 272). This current is undoubtedly an essential feature necessary
for the stability of the tropical current system. Its asymmetry about the equator is a
consequence of the displacement of the thermal equator into the Northern Hemi-
sphere and of the accompanying asymmetry of the atmospheric circulation (see p.463).
The main contributions to the theoretical explanation of the mode of formation of an
Equatorial Counter Current have been primarily due to Sverdrup (1932); Defant
(1935, 1941); Thorade (1941) and Palmen and Montgomery (1940). For an atmos-
pheric circulation assumed symmetrically about the equator, the Equatorial Counter
Current can be readily explained as a compensation current produced by the distur-
bances of the pressure field by a meridional continent opposing the wind drifts
corresponding to the North and South Equatorial Currents. It flows eastwards as a
gradient current in the direction of downward sloping sea level and is retarded only by
friction at the lower boundary surface and at both sides of the current. Stockman
{\9A6a-d) has attempted to consider also the baroclinic mass field, though without
taking into account the dependence of the Coriolis parameter on latitude. According
to this explanation the accumulation of water carried westwards and piled up by the
equatorial currents is the most important factor in the formation of the counter
current. The asymmetry of the counter current about the equator would then be due
to the asymmetry of the atmospheric circulation. Presumably for the Atlantic this
explanation of the counter current can be considered as an adequate one, but for the
considerably more extended Pacific it is doubtful whether the effect of the water accumu-
lation piled up in the west is sufficient in order to give rise to a counter current as a
very narrow band over such a great distance.

The Tropospheric Circulation

599

Evidence against this conception of the equatorial counter current as a pure
gradient current has been accumulated by Sverdrup (1947) and Reid (1948), who
showed that the main features of the baroclinic mass distribution in the tropical and
subtropical Eastern Pacific are due entirely to the effects of the mean wind stress
distribution in these regions. A method for the determination of the mass field and
the mass transport of the currents from the given wind field has already been described
on p. 550 and following pages. By means of Fig. 254 it has been demonstrated that the
mass structure and the currents of the equatorial region of the Eastern Pacific are only
effects of the wind stresses. In these investigations full account was taken of the
dependence of the Coriolis parameter on the latitude, but the influence of lateral
friction and of thermodynamic effects such as radiation and evaporation and others
was neglected. The good agreement between theory and observations is an indication
that the latter effects are of secondary importance in the dynamics of the equatorial
counter current. Figure 273 presents diagrams of forces for the equatorial currents and

Coriolis force

Wind stress

(b)

Wind stress

(c)

Pressure grodient Windstress

Pressure gradient

Pressure yadient

Equatorial

Counter current

Coriolis force

Fig. 273. Diagrams of forces: {a) for the North Equatorial Current; {b) for the South
Equatorial Current; (c) for the Equatorial Counter Current.

for the counter current. Basically there is no difference between them; since they are
each produced and maintained primarily by the wind in a sea with a baroclinic mass
structure.

A comprehensive representation of the oceanic structure and circulation in a section
along the middle axis of the Atlantic is contained schematically in Fig. 274. It is self-
evident that this picture is of a schematic nature only, however, an attempt has been
made to include all the characteristic features of the tropospheric oceanic structure as
well as the corresponding three-dimensional circulatory movements. This circulation
in its zonal extent is largely a consequence of the air currents over the sea surface. The

600

The Tropospheric Circulation

South equ
current

Potar front Convergence
*fE--* — pE— E-T^ ,w-«- *- y-*-' w— H-*^r-^

Po'ar front

Fig. 274. Schematic picture of structure and circulation in the troposphere of Atlantic
Ocean in meridional direction.

Limit between the tropo- and stratosphere.

Position of maximal density gradients.

Tropical-subtropical thermocline.
|^:-!v';v>?.-l Layers of extremely low oxygen contents ( < 1-5 cm^/1).

Position of tropical-subtropical salinity-maxima.

W, E Zonal velocity component (W towards west, E towards east).

meridional components of motion, on the other hand, are a consequence of meri-
dional variations in radiation and evaporation-precipitation difference and are there-
fore only weak.

The lower currents stand clearly out in salinity sections of the Pacific and of the
Indian Ocean as tongues of high salinity. They originate and spread out again from the
subtropical accumulations of highly saline water. A meridional salinity section through
the central part of the Pacific Ocean (Pt. I, p. 172, Fig. 76) shows that the intrusion of
this water from the South Pacific is the stronger one reaching as far as 12° N. in a
depth of 150-250 m. The northern branch, however, is present only between 22° and
25° N. In the east these intrusions seem to be still weaker (see the vertical section in the
Eastern Pacific given by Schott, 1935, p. 182); contrary in the west Pacific region
they are stronger. The southern undercurrent shows as a spectacular phenomenon
(see Fig. 275, Wust, 1929) though again, the northern branch is only weakly devel-
oped.

10° N

Fig. 275. Longitudinal section of salinity through the subtropical deep current in the West
Pacific Ocean (according to Wiist).

The Tropospheric Circulation

601

The equatorial currents are particularly well developed in the Pacific. As in the
Atlantic the counter current lies in the Northern Hemisphere throughout the whole
year and especially far from the equator during the northern summer. The surface
velocities reach values of more than 2 knots. The structure of the water masses was
first pictured in a "Carnegie" section (at about 140° W.) in October 1929 (Sverdrup
et al. 1 942., p. 709). Figure 276 shows the temperature and salinity distributions between

Stot 159
LcrtiO"! S

300

Horizontal velocity, cm/sec

Fig. 276. Temperature, salinity and computed velocity in a vertical section in the Pacific

Ocean between 10° S. and 20° N. (according to the "Carnegie" observations; arrows

indicate direction of the north-south flow; E. and W. indicate flow towards east and west

respectively) (according to Sverdrup, 1942).

the sea surface and 300 m as well as the velocity distribution calculated on the assump-
tion of no motion at the 700-decibar surface. The equatorial counter current hes
between 5° and 10° N., and in correspondence with the sea surface slope flows
downwards in the calm belt between the trade winds. The maximum velocity at
the surface is a little over 50 cm sec-^ in good agreement with observed values. The
"Carnegie" section gives an eastward transport by the equatorial counter current of
approximately 25 million m^ sec-^. The character of the transverse circulation is
evident from the distribution of salinity, oxygen, phosphate and also silicate and is
quite similar to that shown in Fig. 269 derived from observations in the Atlantic.

A detailed theoretical treatment of the circulation in the top layer of the equatorial
parts of the oceans has been given by Yoshida, Mao and Hoover (1953). They start
out with the steady-state equations involving the Coriolis force, the pressure gradient
and horizontal as well as vertical mixing. For the mean wind-stress distribution and

602

The Tropospheric Circulation

the mean density distribution they took Reid's model (1948) which is generally
applicable in equatorial regions. The wind drift and gradient current were super-
imposed correspondingly considering the boundary conditions, and finally the vertical
velocity in the upper mixed layer was calculated using the continuity equation.

Figure 277 shows the dependence on the latitude of the horizontal velocity components
u and V at the sea surface and the horizontal wind stress Tx, acting only in zonal
direction. It is evident that a strong equatorial counter current is formed between

— •(Equatorial Counter Currency

-35 -30 -25 -20 -15

20 25 30

0€ 04 0'2 OO
DYNE/Cm2

Fig. 277. Latitude dependence of the horizontal velocity components u and v at the sea
surface and the horizontal wind stress T^ acting only in zonal direction on the ocean surface.

2-5° and 1 1° N. in the area of weak westward wind stress between the strong north-
east and north-west trade winds. All the velocity components decrease somewhat
with depth down to the lowermost boundary of the upper mixed layer, the w-com-
ponent of the equatorial counter current decreases least so that almost uniform values
are found throughout the entire top layer. The vertical velocity resulting from the
continuity equation is shown in Fig. 278. Its distribution is rather noteworthy. It
shows :

(1) very strong upwelling at or near the equator, this is the equatorial divergence;

(2) strong sinking at the southern boundary of the counter current; and

(3) fairly strong upwelling at the northern boundary of the counter current.

The vertical velocity is of the order of lO-* and 10"^ cm sec"^. Farther to the north
the velocities are small and irregularly distributed. Considering that the Reid model
is only a crude approximation of true conditions and especially that the wind-stress
distribution with zonal components only can hardly correspond to actual conditions,

The Tropospheric Circulation

603

sjstauj U! MidaQ

Depth in meters

604

The Tropospheric Circulation

the similarity with the vertical velocity field shown in Fig. 269 is remarkable. It should
be noted that the vertical velocity component does not vanish at the lower boundary of
the upper mixed layer. The current does not follow the inclined surface of this boun-
dary unless the divergence of mass transport in the upper mixed layer is zero. This does
therefore never correspond to the conditions shown in Fig. 269.

Also in the Indian Ocean conditions are similar with the same much weaker develop-
ment of the phenomenon in its eastern parts (see Pt. I, p. 172, Fig. 75). Since the
thermal equator remains here always south of the equator the tropospheric circula-
tion is again rather asymmetrical and, as in the Pacific, the southern hemispheric
branch is the stronger one. However, while the conditions in this branch are almost
unchanged throughout the total year, complications must appear in the Northern
Hemisphere due to the seasonal changes in the current system of the sea surface.
During the summer months the strong south-west monsoon current extends down to
the lower layers of the troposphere and the subtropical undercurrents are suppressed.
The available sections do not show the nature of this change. Probably the highly
saline water masses of the southern hemispheric lower currents extend into the Nor-
thern Hemisphere and partly enter the influence region of the wind drifts of the south-
west monsoon.

The Equatorial Undercurrent. Cromwell, Montgomery and Stroup (1954)
discovered an Equatorial Undercurrent in the Central Pacific in a zone between the
equator and latitude 1° N. and at a depth of 50-150 m. It is found as a narrow east-
ward current both in the lower part of the top layer at the equator and in the upper
part of the thermocline in this zone, where the South Equatorial Current extends into
the Northern Hemisphere. Its position in the vertical and horizontal circulation of this
area is sketched in Fig. 279. Fofonoff and Montgomery (1955) have shown that the
Equatorial Undercurrent agrees with a simple application of the vorticity equation

E 0 U

ATOR

\ \

EQUATORIAU

CURRENT

"seasuRFace

' eOUATOftua. W«0£RCU«W£hT

Fig. 279. Meridional section showing idealized currents in the surface layer (about 100 m

deep within about 3° of equator, reader looking west). The flux components in the plane

of the section are indicated by broken arrows. Zonal components of velocity at the top

and the bottom of the layer are indicated by diagonal arrows drawn in perspective.

The Tropospheric Circulation 605

(X.68). In a cross-section through the meridional circulation the water flows towards
the equator in the part of the top layer beneath the drift current and rises at the
equator. The zonal component of the surface current can be taken as uniform and the
relative vorticity/o being zero. If a water layer moves without friction from an initial
state ^o> /o' /?o to a new state the vorticity equation gives the relationship

For water sinking from the surface ^q = 0, and if the thickness is assumed to remain
constant during the displacement, and if all the water is assumed to have started from
the same initial state, the distribution of the zonal velocity component can be found
by integration of

f+C=fo- (XIX.2)

For a predominantly zonal current

dii 1 8ii

^ 8y R defy

and for low latitudes the solution can be written in the simpler approximate form

u-Uo = Roj(<j> - cf^of- (XIX.3)

If, in the South Equatorial Current, the surface water has a velocity of 0-5 knots with
no lateral shear and sinks from latitude (f)Q — 2-1° and flows without friction or
changes in thickness to the equator, it will reach the equator as the east undercurrent
with a velocity of 2 knots.

The component of the velocity directed towards the equator in waters moving
from latitude 3^ to the equator can also be calculated. The zonal velocity component
is given by the equation

%=fi-gi.,^ (XIX.4)

where /^.^ is the longitudinal slope of the sea surface at latitude (/>. Its existence is made
possible by the presence of the continental barriers. Since

du dii

fv - -jj ^fv -V ^ ={f+ i)v =foV

(XIX.4) can be written in the form

^ = 7 ix,<i> ~ ^^—r ^x,4>- (XIX. 5)

/o 2cu9o

If ^0 = 3°; /o = 7-6 X 10-*' sec-i and i^^ = — 3 x 10-^ (see Montgomery and
Palmen (1940) and Jerlov (1953)), the velocity component v towards the equator is
—4 cm/sec or 2 nautical miles a day.

The Equatorial Undercurrent is consistent with the flow towards the equator in
the lower part of the top layer close to the equator, if this flow is approximately friction-
less so that absolute cyclonic vorticity is conserved. Continental barriers which permit
a longitudinal component of the pressure gradient, are essential for any extensive
development of the undercurrent.

606

The Tropospheric Circulation

3. Other Currents of the Oceanic Troposhere

(a) The Guiana Current and the Current Conditions of the American Mediterranean

The stream lines of the tropospheric undercurrents of the Southern Hemisphere
converge from the whole of the South Atlantic towards the area off Cape San Roque
on the east coast of South America and the water of the South Equatorial Current
flows into the Northern Hemisphere at this point. The subtropical salinity maximum
of 36-7%o at about 120 m depth can be followed far to the north (as far as the West
Antilles and beyond) in a salinity section following the course of the Guiana Current
north-westwards along the South American coast. The character of this water remains
almost unchanged from the area of South Equatorial Current in the Southern Hemis-
phere to the Antilles. For the most part the current axis remains over the broad shelf
off the mouths of the Amazon and the Orinoco. The corresponding pressure gradient
could be determined so far only from very few stations. The direction of the pressure
gradient in a gradient current must be reversed on passing from the Southern to the
Northern Hemisphere. This can be seen in the sea level topography given in Fig. 271.
South of the equator the higher pressure occurs at the coast with the lower pressure
farther out; north of the equator this is reversed and here the Guiana Current is accom-
panied along its right-hand edge by a narrow ridge of high water level with a down slope
towards the coast which in accord with the great strength of the current is quite
considerable. The Guiana Current, together with the southern part of the North
Equatorial Current, flows into the Caribbean through the passages between the Lesser
Antilles (sill depth less than 1000 m). The observational data for this sea has been
evaluated principally by Parr (1935, 1937«, 1938a); see also Seiwell (1938) and
Rakestraw and Smith (1937) on chemical aspects and a review of these conditions
by Dietrich (1939). The tropospheric currents between 100 and 200 m are very
clearly shown by the salinity maxima of the undercurrents which are a continuation of
those of the North and South Equatorial Currents. Figure 280 shows a chart of surface

Fig. 280. Distribution of salinity in the core of the subtropical salinity maximum in the
American Mediterranean (according to Dietrich).

The Tropospheric Circulation 607

currents during the spring. The large salinity diflferences which appear where the under-
currents of the Equatorial Current join off the Antilles soon disappear in the eastern
Carribean. There is a striking uniformity in the Caribbean and in the Yucatan Channel
due to lateral mixing. The weak inflow through the Windward Passage (sill depth
about 1600 m) makes little change. The differences in the Gulf of Mexico are larger.
The extended areas with vortices in the north-eastern and the western parts of the Gulf
which are very pronounced in the surface currents remain outside the circulation of
the tropospheric layers. Investigation of [r,5']-curves in the water masses of the South
Equatorial Current, the Sargasso Sea and the Yucatan Channel allows to estimate
how much of the inflow water through the Antilles takes part in the water passing
through the Yucatan Channel. Iselin (1936) found that of a total transport of about
26 million m^/sec through the Yucatan channel approximately 6 million originates in
the South Atlantic. For the deeper layers the eff'ects of the inflow through the Wind-
ward Passage and the Virgin Passage are of greater importance (see Pt. I, p. 133).
The uniformity of the distribution of the oceanographic factors over the area shows
the effect of the mixing processes which are stronger here than the pure transport
processes. Dynamic evaluation of the data for the latter should give greater informa-
tion (Parr 1937Z?). Figure 281 gives the dynamic topography of the physical sea level
relative to that of the 1200-decibar surface for the Caribbean and for the Cayman
Sea. The mean current core running from the Antilles through the Yucatan Strait to
the Florida Strait is clearly marked. The course of the dynamic isobaths shows that
the water flows uphill to reach the Yucatan Channel (see also Sverdrup, 1939).
Similarly as in both the Equatorial Currents, the water transport here is also largely
due to the air currents (prevailing wind to the east-north-east with a mean velocity
of 10 m/sec). Thus to a very large extent these currents are also gradient currents in a
baroclinic sea though they are subject to significant modification by the wind.

{b) The Gulf Stream and its Internal Structure

Although the Gulf Stream is the largest and the most important current of the
Northern Hemisphere a more dynamic investigation of its course has only recently
been started. The first current measurements in it were made by Pillsbury in 1885-9
from the "Blake" which was anchored in very deep water. Further investigations
were begun in 1914 by the oceanographic survey vessel "Bache" (Bigelow, 1917,
four transverse profiles through the Florida Current and the Antilles Current). More
recently a systematic survey has been started by the oceanographic survey vessel
"Atlantis" (Woods Hole Oceanographic Institution).

The first dynamical evaluation of some transverse profiles in the Florida Current
was given by WtJST (1924) using the "Blake" measurements. This and subsequent work
have aff'orded a more or less complete description of the vertical structure of this
current from the Florida Strait to the Newfoundland Banks. Special mention should
be made of the work of Jacobsen (1929) on the Sargasso Sea using "Dana" observa-
tions and that of Iselin (1936) giving a detailed review of the comprehensive results
collected by "Atlantis". Dietrich {\92>lb, see also WiJST, 1930a) has given a detailed
analysis of numerous sections to show the process of formation and the dynamics of
the Gulf Stream. The thermo-haline structure of the Gulf Stream is immediately
apparent from the set of six success profiles given by WiJST (Figs. 282, 283). Profile I

608

The Tropospheric Circulation

85° VV ^0° 75° 70°

85° W

Fig. 281. Dynamic topography of the physical sea surface (relative to that of the 1200

decibar surface) for the Caribian Sea and the Cayman Sea. (Lines of equal dynamic anomaly

drawn with an interval of 005 dyn m.)

is in the Yucatan Channel, profile II north of Cuba, profile XII at the narrowest
part of the Florida Strait, profile IV at the exit from the Florida Strait just before the
junction with the Antilles Current, profile V at Cape Hatteras and profile VI from the
Newfoundland Banks in southward direction. The temperature profiles show that the
Gulf Stream is by no means a deep-reaching current of high temperature. It differs
only little in the thermal structure from the neighbouring Sargasso Sea. The steep
oblique slope of the isothermals and isohalines is characteristic and the narrower the
section the more rises the lower, cold and weakly saline water at the left-hand boundary.
This baroclinic mass distribution is connected with the current velocity and direction
and is more pronounced the stronger the flow. It is thus more prominent in the narrow
sections to the south. Profile V shows the Gulf Stream beyond the junction of the
Florida Current and the Antilles Current where it has its greatest vertical thickness,
about 1000 m, and has the considerable core width of about 50-70 km. Its left-hand

The Tropospheric Circulation

609

New Foundtand Bonk

Sargasso Sea

Fig. 282. Cross-section of temperature through the Gulf Stream (profiles I, lla and V
according to Jacobsen; profile VI according to Helland-Hansen; profile II and IV according

to Wiist).

610

The Twpospheric Circulation

New Foundlond Bank

rrhernportof^x <;, AvVV"^.".

Sorgasso Sea

100 200 km 300

Fig. 283. Cross-section of salinity through the Gulf Stream (see remarks below Fig. 282).

The Tropospheric Circulation 611

edge is sharply defined and keeps about 100 km off the coast. The right-hand edge is
diffuse and differs little from the water farther to the east. Where it swings eastward
the current spreads out and loses thermal and haline thickness by mixing with colder
surrounding waters. To the south of the Newfoundland Banks it begins to break up
into a number of branches ; profile VI shows only the northern branch which borders
on the Labrador Current. The further branching of the current in the east has been
discussed on p. 562, Fig. 257.

The Gulf Stream is only slightly more saline than the Sargasso Sea and in the
deep layers there is no difference. The salinity maximum lies in the subtropical
undercurrents which enter through the Antilles as part of the North and South
Equatorial Currents into the American Mediterranean and from there across the Gulf
of Mexico into the Florida Strait. It is thus a long-range effect of the tropospheric
circulation of the tropical and subtropical Atlantic. The Antilles Current also shows
this highly saline intermediate layer; but here it is in direct connection with the highly
saline top layer of the Sargasso Sea. Further along the course of the Gulf Stream this
salinity maximum comes at times up to the surface, but in the North Atlantic Current
it dips beneath the weakly saline surface layers. It can be traced well into the Norwegian
Sea (see pt. I, p. 171, Fig. 74). The salinity profile also shows another long-range effect
of the Atlantic circulation: this is the last traces of the weakly saline subantarctic
intermediate water which can still be seen at a depth of between 700 and 1000 m
{S < 34-9%o) as far north as 25° N. in the Florida Strait; in the Sargasso Sea, however,
it reaches only to 10° N.

The dynamics and the water transport of the Gulf Stream are derived primarily
from velocity profiles. Several such profiles are available at the present time; they are
based partly on direct-current measurements and partly on dynamic calculations from
the mass field. The cross-section is not everywhere completely occupied by the current;
particularly where the current flows out of the Florida Strait into the open ocean.
The current flows as a jet through the narrow part of the strait and follows the direction
imposed on it for a considerable distance. The velocity distribution in the cross-
section is related to the mass field and the agreement between the calculated and
observed current profiles is generally good. Beyond the junction of the Florida
Current and the Antilles Current the weak counter current between them disappears
completely, but the counter current on the right-hand side of the main one is retained.
In the cross-section off Chesapeake Bay shown in Fig. 284 it lies just outside the
profile.

A deeper insight into the dynamics of the current can be obtained from the absolute
topography of the isobaric surfaces and of the physical sea level. These are parti-
cularly dependent on the choice of the reference-level. In the narrows of the Florida
Strait this lies near the bottom where the velocity decreases almost to zero. Further
north it lies in the Sargasso Sea at about 1900 m depth (corresponding to Fig. 272) and
rises steeply from the right-hand side of the Gulf Stream to 1000 m depth or even less.
Over the current core the physical sea level rises steeply from left to right and at Cape
Hatteras this rise amounts to about 100 dyn cm. It remains more or less of the same
order up to the Newfoundland Banks, but gradually spreads out horizontally so that
the actual gradient falls to about a third. The right-hand side of the Gulf Stream is
associated with a high-pressure ridge which can be traced from the Bahamas to the

612

The Tropospheric Circulation

Coostal stream Gulf stream

Fig. 284, Velocity profile (cm/sec) across the Gulf Stream off Chesapeake Bay, 20-22 April

1932.

south-west of the Newfoundland Banks. Eastwards from here there is a counter
current steadily broadening to the south. The absolute topography of the 500 decibar
surface still shows clearly the same pressure gradient as at the sea surface but it is
rather weakened. The 800 decibar surface shows a rise across the current of at the
most 20 dyn cm; and the pressure gradient has fallen to about a quarter. The 1400
decibar surface is almost plane and the lower limit of the current system must there-
fore lie between 1000 and 1200 m.

A detailed analysis of the origin and the transformations of the Gulf Stream water
as it flows from the Florida Strait to the Newfoundland Banks were investigated by
Dietrich (1937) with the aid of distribution of oxygen content in numerous profiles.
He was able to show that the water masses of the Florida Strait and of the Antilles
Current to the north of the Lesser Antilles were made up partly of tropical South
Atlantic water and partly of subtropical water from the western North Atlantic. The
Gulf Stream water reaching Cape Hatteras has, however, undergone changes making it
almost completely identical in its properties with the water of the western North
Atlantic. This transformation was attributed by Dietrich to the transverse circulation
and to mixing. From the distribution of the oceanographic factors such a transverse
circulation seems not unlikely, but it is not possible to determine it from the pressure
field because of the low velocity and probably also because of its variability.

The amounts of water and heat carried by the Gulf Stream are enormous. The
Florida section shown in Fig. 284 gives a water transport of about 25 million m^sec.
It can be assumed that this will also be the transport in the currents through the Carib-
bean and the Yucatan Channel, since the precipitation and the inflow of river water
(run-off) are small compared with this very large quantity. Some idea of the enormous
quantity of water involved is given by the estimate that it is twenty-two times as much
as is carried by all the rivers of the earth together. The amount of water carried by the
Gulf Stream further north is much larger than this and the transverse profile off
Chesapeake Bay gives a transport three times greater (82 million m^sec). It can be
assumed as a first approximation that the amount of water carried by the North

The Tropospheric Circulation

613

Equatorial Current, together with that carried by the Guiana Current and passing
between the Lesser Antilles will be about the same as the total transport of the Gulf
Stream through a cross-section off Cape Hatteras. From this it follows that the part
of the Gulf Stream that passes through the Florida Strait makes up only about a
third of the total transport. According to WiJST the Antilles Current carries 12 million
mVsec and the Florida Current about 37 million m^/sec. From here the current enters
regions with larger depth and there occurs a rapid increase in the water transport
because the current absorbs water masses with a temperature of less than 8 ° C from
the lower layers of the south-western Sargasso Sea. Further along to the north and
north-east the Gulf Stream is subject to a velocity decrease and an increase in width,
but the water transport remains nearly constant. However, it becomes more and more
difficult to distinguish its limits from the surrounding sea. Iselin has attempted to
divide up the Gulf Stream velocity profile at Chesapeake Bay (Fig. 284) into individual
inflow components (Fig. 285). The area A contains water warmer than 20° C and the

200
400
600
800
1000
1200
1400
1600
1800

Fig. 285. Subdivisions of the velocity profile across the Gulf Stream off Chesapeake Bay,

20-22 April 1932. The figures give the transport (in mill, m^ sec"^) for the different parts of

the current (according to Iselin).

velocity of this gives a transport of 10-6 million m^sec. The same layer in the Florida
current according to the WUst profile corresponds to 13-1 million m^sec and in the
Antilles Current to 4 million m^sec. The sum of these two is greater but no more so
than could be due to differences in the homogenity of the material. The area B contains
only water colder than 8° C, most of which was absorbed by the Gulf Stream in the
section with a larger depth. According to the velocity profile tliis area corresponds to
12-7 million m^/sec and only a very small part of it can possibly be assumed to have its
origin in the Florida Strait. Water is also drawn into the main current along both
edges by friction and mixing. If these areas in the profile are limited by the isoline of
20 cm/sec, these areas C and D will correspond to a transport of 0-7 and 12T million
m^sec respectively. These figures indicate that water is drawn into the current on the
right-hand side much more strongly than along the more sharply defined left-hand
boundary. The remaining area E corresponds to 46-1 million m^/sec. In the Wiist
profile for the Florida Current and the Antilles Current this area corresponds to

614

The Tropospheric Circulation

about 26-4 million m^sec. The transport in the current core has thus grown to twice
its magnitude in a distance of about 600 nautical miles. This very large increase from
26 to 83 million m^sec, where the current passes into a region with larger depth, can
be attributed to three principal sources. The smallest of these is due to the Antilles
Current which brings the total transport up to 37-1 million m^sec leaving 45 million
to be accounted for from the other sources. This is supplied, on the one hand, by water
drawn in from the south-western part of the Sargasso Sea and on the other hand, by
water fed by the counter current coming from the area of the Newfoundland Banks
and mixed with the Gulf Stream by means of numerous vortices. In this way Iselin
derived the schematic outline of the main sources and of the course of the Gulf Stream
shown in Fig. 286. Each line represents a water transport of about 12 million m^sec.
This may seem somewhat schematic, however, it gives an instructive idea about the
origin and composition of the water masses transported by the Gulf Stream.

80° 70 60 50 4

0 20 10° W

Fig. 286. Schematic representation of the main sources of the Gulf Stream waters (broken
lines) and the pattern of the Gulf Stream system (continuous lines). In the western half of the
ocean each stream line represents a water transport of approximately 12 x 10® m^ sec~^

(according to Iselin, 1935).

The systematic survey of the Gulf Stream between Montauk Point and the Bermudas
carried out by the "Atlantis" from June 1937 (Iselin, 1940) showed that the mass
transport of the Gulf Stream varied between 93 and 76 million m^sec. There was a
definite annual variation with two maxima in early summer and in winter and two
minima in October-November and April-May. The differences in the sea-level across
the current are closely related to these variations and can be deduced from them.
This annual variation is probably due to variations in the intensity of the atmospheric

The Tropospheric Circulation 615

circulation over the southern part of the North Atlantic. In winter the strong anti-
cyclonic circulation over the ocean increases the inflow into the Gulf Stream and in the
summer there are more frequent southerly winds and a greater part of the water masses
of the North Equatorial Current is blown directly into the Gulf Stream without passing
through the Carribean and the Florida Strait. Both of these effects intensify the Gulf
Stream. It is not improbable that the aperiodic variations from year to year wil
provide an extremely good indicator of the variations in the intensity of the atmos-
pheric circulation over the Atlantic.

The most recent investigations of the Gulf Stream have the main goal to obtain
accurate detailed surveys of the current at short successive intervals, that is, to obtain
quasi-synoptic surveys of an extended part of the current. Such methods of investiga-
tion need in the first place the rapid gain of the structure of the water masses down to
great depths, while the survey vessel is under way whereby the position of each station
has to be fixed with accuracy. Both of these conditions can be satisfied by the more
recent methods used on board of the oceanographic survey vessels. Quasi-synoptic
surveys of this type have been made for the Gulf Stream down to 275 m depth between
Cape Hatteras and the Newfoundland Banks but at the present time only few of them
exist. They give a very clear picture of the complicated course of the current and show
particularly the very considerable local variations in form of meandering wave patterns
of large amplitude at both sides of the current. Occasionally a water mass in one of the
amplified troughs and ridges is cut off from the main current to form finally a large
vortex which will be cyclonic on the southern side and anticyclonic on the northern
side. These vortices are different from the smaller size eddies in the shearing zones of a
turbulent current that also occur in the Gulf Stream (Spilhaus, 1940). Furthermore,
the synoptic surveys have shown that the current velocity in the core may be intensified
up to about 4-5 knots over a relatively narrow band (about 10-15 miles wide) a little
inside the left-hand boundary of the current; in the counter current the velocity reaches
3-4 knots. It is not surprising that the approximate and mean values obtained by the
previous methods of investigation gave only low velocities.

The first multiple ship survey of the Gulf Stream area between Cape Hatteras and
the Newfoundland Banks was made during 6-23 June 1950. Six oceanographic survey
vessels took part in this "Operation Cabot" and they obtained an almost synoptic
survey of the Gulf Stream down to 275 m which gave a clear picture of the compli-
cated nature of the current. Figure 287 presents the course of the current as character-
ized by the mean temperature of the upper 200 m layer. According to this survey the
Gulf Stream is a remarkably narrow band about 40-60 km wide and sharply separated
at the edges from the surrounding water masses. The early view of Franklin of the Gulf
Stream structure was confirmed, and certainly in the sector between Cape Hatteras
and the Newfoundland Banks the Gulf Stream resembles a "river in the ocean"
rather than a broad diffuse ocean current. The current does not, however, follow a
straight line, but instead flows in long waves which are usually of small amplitude but
take occasionally quite a large amplitude. Successive surveys have shown that these
long lateral waves move slowly eastwards with increasing amplitude. Figure 288 shows
the position of the Gulf Stream at the beginning (8 June) and the end of the operation
21 and 22 June). The current core therefore tends towards a meandering behaviour
of a pronounced character. The amplitude of these meanders may increase so much

616

77?^ Tropospheric Circulation

66°

55°

5,„. ^^^

^°°M§^^

"s=

^^^

^glls

J-

"'"'^-V

'»^ ^

'■,,-73°'

n>>

.... \,

72°

74° J
'73°

MJ J

::;,--S?

*^'"

Ships t racks -~-,,,^--

ULK Current airecTions -♦■

36"

38»

740 730 72° 71° 70° 69° 68° 67° 66° 65°

Fig. 287. Mean temperature (°F) in the upper 200 m layer of the Gulf Stream, 8 June 1950.

while moving eastwards that large sections of the current can be cut off. This process
results in the ejection of a water mass from the current and the formation of large
cyclonic vortices on the southern side of the main current. This cut-off process is
similar to processes involved in polar jet phenomena in the upper atmosphere which
are of major importance in the dynamics of these air currents. The process can be
followed clearly in successive charts from 16 June to 21 June. On 17 June this process
reaches its maximum stage (Fig. 289). The cyclonic vortex clearly stands out in the
band of temperature concentration and in direct current recordings. It was at first a
strong vortex but gradually weakened during the following days and finally vanished.

A further characteristic phenomenon is the break-up of the Gulf Stream into
several separate branches. Usually there are three, sometimes separated by counter
currents. Figure 290 shows the current velocity and temperature distribution usually
found at the sea surface.

Consideration of these recent results shows that there are three principal questions
on the internal dynamics of the Gulf Stream that require an answer.

(1) Why is the current asymmetrically developed and why is the current core
displaced to the left-hand side (looking downstream) ?

,72° 71° 70° 69° W 68° 67° 66° 65°

Fig. 288. Position of the Gulf Stream. Mean temperature (°F) of the upper 200 m layer for
8 June (full lines) and for 21 and 22 June 1950 (dashed lines).

The Tropospheric Circulation

617

Fig. 289. Mean temperature (°F) in the upper 200 m layer on 17 June 1950. Current
direction from geomagnetic electrokinetograph (GEK) (according to Arx, 1950).

(2) Why does the Gulf Stream keep such a concentrated narrow form over a long
distance sometimes taking on a meandering character? Why does it break up into
several smaller branches separated by motionless bands or weak counter currents?

(3) Why is the total energy of the current concentrated in a relatively thin top layer
and why does the current not penetrate down to the deeper layers when it flows out
over regions with larger depth?

Research on these questions is in progress but more fundamental results have been
obtained only for some individual questions. It appears that these strong oceanic
boundary currents are analogous in many respects to the "jet streams" of the strong
westerlies in the upper atmosphere and are especially characteristic for the dynamics
of free jets.

RossBY (1936, 1937, 1938) in a series of papers has advanced some new ideas on the
theory of ocean currents which are of some interest. These arguments have been
applied primarily to the Gulf Stream System between the Florida Strait and the area
south of Newfoundland. But their use is not limited to these currents and in many
respects they can also be appHed to all boundary currents flowing parallel to a coast

618

The Tropospheric Circulation

The Tropospheric Circulation 619

(Kuroshio, Peru Current) and others. Rossby's theoretical investigations are put
forward mainly along two lines. The first deals besides the vertical also with the lateral
frictional effect which is of influence on the horizontal velocity profile in currents.
The second deals with currents of constant momentum (impulse) transport and in
particular applies the theory of free jets to ocean currents. Since the exchange co-
efficients of lateral mixing are of considerably greater magnitude than those for vertical
exchange (see Pt. I, p. 103 et seq.) Rossby considered it absolutly necessary to account
for frictional forces due to lateral mixing and put strong emphasis on these forces.
The usual equilibrium conditions in a geostrophic current for mass elements along a
vertical line primarily determine — as always^the vertical velocity distribution. By
introduction of the lateral shearing forces this condition will not be changed in any
great extent, but the lateral shear imposes a definite transverse velocity profile to which
little attention has been paid in the past.

A linear current in the positive v-direction with a mean velocity v will be fixed by
the geostrophic equilibrium between the pressure gradient —(1/ p)(dp/8x) and the
Coriolis term —fv. As a result of the horizontal turbulence, however, the individual
mass elements will have a movement at right angles to the mean direction of the current
and the equations of motion (XIII. 1), will apply for its horizontal components u and v.
If the deviations of w and v from the mean velocities « = 0 and I; are denoted by
u' and v' then:

dv' ^ , du' 1 dp

T,=-^" and ^=A.' w,th -^£-/S = 0. (XIX.6)

From (X.39) the lateral shearing stress is

T = - pTlT. (XIX.7)

Introducing the Prandtl mixing length / of lateral mixing (p. 388) allows (XIX.7) to
be rewritten as

= p/«' (/+ ^)- (XIX-8)

For a uniform horizontal current the lateral shearing stress will not approximate to
zero except when

Under stationary conditions the lateral mixing imposes a definite horizontal velocity
profile, and indeed there must be a velocity decrease towards the right-hand edge of
the current (Northern Hemisphere). This is quite large and in middle latitudes (43°)
amounts to 1 cm/sec in 100 m).*

Since such large transverse variations in velocity are hard to observe it must be
presumed that the right-hand edge of the current always tends to accelerate the left-
hand side even when the right-hand side has a lower velocity. This effect ceases only
when the condition (XIX. 9) is satisfied.

* Against this conclusion the objection has been raised by Priebsch (1943), that besides the
lateral turbulence across the gradient current, also that in the direction of the current should be
taken into account. If this is done, it is found that the effect ot the earth's rotation mentioned
above no longer exists. On the average the effects in the two directions balance exactly.

620 The Tropospheric Circulation

The second part of Rossby's arguments concerns the problem of a straight accel-
erated turbulent current. In such a current the horizontal pressure gradient will not be
equal to that corresponding to the meari basic current and not balance completely
the Coriolis force in stationary equilibrium. This gradient of the stationary current
was termed the Coriolis pressure gradient by Rossby and for this the following relations
apply

-^^-p/i; and -^^ ^ + pfu. (XIX.IO)

A numerical value can always be found for given u and r. The turbulent accelerated
motion, however, will be subject to other equations:

du dp St„„

and dv ^ ^P , ^'^vx

where r^y and Ty^ are the x- and jF-components of the lateral shearing stress. Intro-
ducing p = Pc -\- Pr then by means of (XIX.IO)

and dv dp,. dxy^

^dt^~d^'^ '8^'

The movements which correspond to these equation occur under influence of
"residual pressure gradients" Pr as though the earth was not rotating. The continuity
equation

du dv

ai + a7 = ° *^"''"

fixes the current field u, v, while (XIX.IO) gives the Coriolis pressure gradient and the
corresponding mass field. Since /?(. is usually considerably greater thanpr it is clear that
the general pressure distribution is of secondary importance in considering accelerated
currents. The mass field which is determined by the mean steady current field gives
no information on the cause of the currents. However, according to Rossby /j^ should,
dynamically, be more important than p^.

Against these considerations Defant (1937) and Ekman (1939) have raised doubts
affecting more particularly the practical usefulness of the above equations. But
nothing can be said generally against the main lines of the basic argument if one
remains in agreement with actual conditions.

For an application of the above equations to Gulf Stream problems Rossby took
into account the phenomena that occur when the flow of a medium takes the form of
a jet. The theory of free jets (Prandtl, 1926) has been further developed by Tollmien,
1926; FoRTHMAN, 1934; Ruden, 1933. For a steady state (du/dt = 0) in a laterally
restricted current the first of the equations (XIX. 12) together with the continuity equa-
tion (XIX. 13) gives

pu^ dy = constant, (XIX. 14)

The Tropospheric Circulation

621

that is, in a current of this type the momentum {impulse) transport through a current
cross-section is constant. Neglecting dp^jcx (which is permissible) and introducing
the shearing given by

'»-'fy

according to equation (XII. 15), then for a mixing length / = ex (proportional to the
distance travelled) a complete solution can be found that fixes the horizontal current
profile in the free jet. The very good agreement between theory and experimental
results for the current profile in a free jet, is a consequence of the assumption made for
the mixing length which is completely valid only for limited dimensions. Whether it is
also applicable for the very large dimensions of ocean currents is questionable.

One consequence of the assumption is also that in a free jet with constant momen-
tum transport the mass transport increases downstream, and is in fact proportional to
the square root of the distance travelled. Due to the incorporation of surrounding
water the current cross-section will increase downstream while the mean velocity will
decrease. Since the energy remains the same, the mass transport will increase. Condi-
tions are somewhat different if the inflow through the initial cross-section does not
start from a point source but has a finite width. The velocity profile in Fig. 291 is

15
\-0-\

0-5

1-0-
1-5-

FiG. 291. Velocity distributions in a jet (Freistrahl, according to Ruden). D, nozzle diameter,
all lengths are given as multiples of D.

based on experimental values for the velocity at different distances from the outlet of
a nozzle through which there is a constant inflow. In a free jet there is a core in which
the initial velocity and the other properties of the medium remain unchanged for a
relatively long distance from the nozzle. The formation of a core region and a surroun-
ding one of mixing are characteristic of the phenomena occurring in the ocean under
similar conditions.

These results apply in the absence of rotation. According to Rossby, the principal
effect of earth rotation is the formation of a different mass distribution (according to
(XIX. 10)) corresponding to the Coriolis pressure force; the velocity profile, however,
will not be disturbed further by it. The stationary properties of the current, that is that
the stream lines, isobars and contours of the physical sea level coincide, remain more
or less unchanged. The deviations from a geostrophic current occurring in the interior
of the free jet that are produced by the shearing stress, will be accentuated by the
deviations due to inertia. There will thus be an overall dynamic equilibrium. According

622

The Tropospheric Circulation

to this concept it is the residual pressure field even though it is weak that provides the
driving forces. This is the basic idea of the Rossby theory. It is undoubtedly attractive
but whether it actually corresponds to reality is impossible to say. In any case it
deserves considerable attention.

The further phenomena that occur when the medium in which the free jet is formed,
is stratified, can be fairly readily dealt with. If there are two layers in the medium the
velocity of the upper layer will affect the sea surface slope and also the position of the
internal boundary surface between the two water masses. The sea surface slope and
the internal boundary slope are given by equations (XIV. 6 and 7) (p. 455). If the lower
layer is assumed to be motionless then the velocity of the free jet gives the mass distri-
bution in a transverse section. This is shown schematically in Fig. 292. In the current the

Motionless 1 Jet current
500

a 1000

1500

Motionless

ibb'^>.^

Fig. 292. Cross-section through a jet (Freistrahl) current in a two-layered ocean.

boundary surface will slope downwards, in the Northern Hemisphere from left to
right, and the thickness of the free jet will therefore vary across the current. The total
mass transport through a cross section will be

M = lpu{D, + Ci + y dy.

(XIX. 15)

where Dq is the mean thickness of the top layer and ^^ and i^ are the deviations of the
physical sea level and the boundary surface from their positions when the system is at
rest. Evaluation of this integral gives the result that the difference. Drigiu — ^left
between the two sides of the current must increase downstream as long as the mass
transport increases. This has several effects on the course of the current. The inflow
of water from the surroundings into the free jet will be asymmetric because of the
asymmetry of the system. On the left there will be only a shallow water layer available,
but on the right the water can be drawn in from greater depths. Under steady conditions
the transverse velocity must therefore be greater on the left-hand side than on the
right.

The surrounding water masses can be assumed to be stationary, but this state can
hardly be expected to persist under the given conditions. At some distance from the
current boundary the thickness of the layer D in the motionless water will be somewhat
greater than Dieit and Dri^ht at the left- and right-hand edges. On the way from motion-
less water towards the boundary and into the interior of the free jet the water columns
drawn into the current will undergo deformations, which will be associated with
hydrodynamic vortex formation at the current boundary. The theoretical form for a
cross-section through a free jet of this type is that shown in Fig. 293. This requires

The Tropospheric Circulation

623

a counter current at the left-hand edge of the free jet and a current in the direction of the
main current at the right-hand edge. Due to the increased sea level difference between
the right- and left-hand sides, the counter current on the left-hand side will increase in
strength downstream, while on the contrary the other current will become weaker
on the right-hand side until it finally vanishes. The effect of the free jet and the counter
current must thus increase steadily downstream and therefore tend towards impossible
unstable conditions. The effect of the vortex formation will give rise to a water move-
ment through the main body of the current from right to left, and since the left-hand

500

1000

1500

Deep woter
(motionless)

Fig. 293. Cross-section through a jet (Freistrahl) current in a two-layered ocean with a full
development of a counter current and compensation current in the adjacent water masses

(according to Rossby).

edge and the counter current are shallow there must also be a transverse current in
the lower part of the top layer in the opposite direction in order to compensate the
upper transport. This gives a cross-circulation as was assumed by Dietrich. In addition
to his earlier work, the processes occurring at the edges of a jet-form current penetra-
ting a motionless water body have been discussed in two later papers by Rossby
(1937, 1938). Thereby, he assumed that the initiation of the current from a state of
rest was due to a wind field whose action was restricted to a band-like oceanic region.
Particular attention was paid on the one hand to processes at the edges of the current,
on the other hand to oscillatory processes which occur while the current tends towards
a steady state. In such cases counter currents are formed on both sides of the basic
current; in homogeneous water they are broad and slow, but in a two-layered sea
narrow and intense. The zones between the basic current and the counter current are
dynamically unstable and show a tendency to break up into large horizontal vortices.
The depth to which a surface disturbance may penetrate into the lower layer down to
the sea bottom, and the time required for the restoration of stationary conditions, are
of particular interest and are especially important in dynamic oceanography (see
Chap. XXI. 4).

Without question the theory has applications to the Gulf Stream between the Florida
Strait and the Newfoundland Banks, and several theoretical consequences are un-
doubtedly realized in the actual behaviour of the Gulf Stream. The criticism on this
theory expressed by Ekman is concerned not so much with the theoretical fundaments,
but more with the question of the extent to which the Gulf Stream actually keeps the
character of a free jet and contains the energy (momentum of motion) required by the

624

The Tropospheric Circulation

theory. By means of approximate calculations he was able to demonstrate that the
current leaving the Florida Strait will probably have a kinetic energy, so that already
half of this energy would be able to carry the water against frictional resistances of
various types exactly as far as a wind of 3 to 4 Beaufort could do blowing from the
Florida Strait until Cape Hatteras in the current direction. Over this section of the
current the theory should be able to make the most important characteristics of the
Gulf Stream understandable. However, for the section of the current from 60° to 20° W
conditions appear to be rather different, and in this section the initial velocity of the
water seems to be only of minor significance. Ekman therefore came to the conclusion
that for most of the ocean currents the theory is of limited usefulness only and can
be applied solely to very fast currents (such as the Florida Current and its immediate
continuation, see also, Thorade, 1938). The Rossby theory, due to its consequent
and careful style, had a very stimulating effect and has lead to a better understanding
of a number of phenomena displayed by the Gulf Stream between Cape Hatteras and
the Newfoundland Banks.

A satisfactory theory of the Gulf Stream must take into account a further important
fact that has already been referred to by Dietrich (1937a). Determination of the mean
sea level along the North American coast from Florida in the south to Nova Scotia
in the north by means of precise trigonometric measurements has shown that the sea
level rises along this total route to the north with a mean slope of 13 cm in 1000 km.
The strongest slope occurs just north of Cape Hatteras (see Table 152 according to
An VERS, 1927 and Rappleye, 1932).

The Gulf Stream thus shows an w/7H'(7r(/ motion along this section like the Caribbean
Current where according to Parr and Sverdrup (p. 607) there is a slope of about the
same magnitude. However, as it was shown by Dietrich, that the Gulf Stream in
contrast to the Caribbean Current does not show this slope when the physical sea level

Table 152. Average Mean Water Along the North American East Coast.
(Zero point relative to Florida-Georgia)

Location

Mean water

Distance
along the
coast (km)

Anvers

Rappleye

Average

per 1000 km

St Augustine, Fla.
Femandina, Fla.
Brunswick, Ga.

]

Norfolk, Va.

1000

Cape May, N.J.
Atlantic City, N.J.
Fort Hamilton, N.J.

1400

35
13

Boston, Mass.
Portland, Me.

}

2000

Halifax, Nova Scotia

2600

topography is calculated from the mass distribution along the continental slope.
Dietrich took the oxygen minimum layer as reference-level but recalculation for a

The Tropospheric Circulation

625

deeper reference level changes the results very little. There is thus a contradiction
between the "geodetic" and the "oceanographic" levelling which requires explanation.
A plausible explanation was indicated by Sverdrup (and co-workers 1946, p. 578)
based on the following assumptions.

(1) That the geodetically determined gradient of the sea level is actually present in
the coastal waters just off the coast and that corresponding to this there is a coastal
current flowing southwards.

(2) That in the neighbouring waters the physical sea level slopes down seawards
until the left-hand edge of the Gulf Stream which causes a current to flow southward
due to the piling up of water. This gradient current would be one part of the large
elongated vortex on the left-hand side of the Gulf Stream while the second part flows
along the left-hand edge of the main current and in the same direction.

(3) Corresponding to this current and the adjoining Gulf Stream, the physical sea
level rises steeply seaward from the coast (p. 607). The depression showing the deepest
water level thus would follow the continental slope rather closely so that the south-
ward flowing branch of the vortex lies over the shelf. The topography in a transverse
section across the Gulf Stream thus has some similarity with that shown in Fig. 203

1231 1230 1229 1228 1227

1226

500

1000

1500

2000

„Atlontis"
_ April 1932

Fig. 294. Density distribution (at) and position of the lower limit d of the current system
in a cross-section through the Gulf Stream. "Atlantis", April 1932 (Chesapeake Bay,

Bermuda).

(p. 460) south of the Newfoundland Banks where on the other side of the depression
in sea level the Labrador Current flows eastward. According to the Rossby theory the
elongated vortex between the Gulf Stream and the continental slope is a dynamic
necessity. The Gulf Stream now would flow downhill in accordance with its mass
structure and the surface slope would be directed southwards only at the coast. This
piling up of water over the continental shelf was regarded by Sverdrup as due to the
prevailing wind over the North Atlantic. The south-west wind over the northern part
of this ocean maintains a high water level along its northern borders and maintains

626

The Tropospheric Circulation

in this way a decline of the physical sea level along the eastern and western sides. This
would be the geodetically determined rise between Florida and Nova Scotia.

All cross-sections through the Gulf Stream show a strong stratification in the upper
layers but beneath this where the current is weak it is less pronounced. It is to be
expected that there will be a layer of no motion just beneath this layer. Figure 294 given
by Neumann (1956) shows the position of the zero level d in an "Atlantic" cross-
section through the Gulf Stream. The latter one indicates clearly the form given in
Fig. 292 with shallow depth along the left-hand edge of the current, a strong down-
ward slope below the maximum transverse density change and uniform larger values
at the right-hand edge. Tliis distribution is characteristic of all sections through free
jet currents in the ocean. Neumann has also shown that over the whole of the moving
layer from the surface down to the depth of no motion d there are only slight changes
in the mean density distribution. There exists thus in a first approximation no trans-
verse density gradient. This means that the entire current system is an equivalent to
that of a two-layered model in which there are two water bodies, one on top of the
other with an internal boundary surface between. Thus as a first approximation the
Gulf Stream can be regarded as an equivalent-barotropic system in which the boundary
layer slopes downward from the left-hand to the right-hand edge. Figure 295 shows the

1231 1230 1229 1228 1227

1226

500

1000

1500

2000

50km

Fig. 295. Velocity distribution in the "Atlantis", section Chesapeake Bay-Bermuda, April
1932 {d, lower limit of the current system).

velocity distribution calculated from the mass field (Fig. 295) for a cross-section at the
lower limit of the current system d. For the vertical shear under equivalent-barotropic
conditions as a first approximation one obtains

(XIX. 16)

dv g dp

dz f p 8x'

where p is the mean density of the current layer. If as on p. 608 t, denotes the surface

The Tropospheric Circulation

eii

of the physical sea level and —d and p_a are the depth and the corresponding density
of the layer of no motion, then from (XIX. 16) when m_<j = 0 and

it follows

Vr =

P =

l + d

If dpjcx is exactly zero then

Vr =

- P-d ^ I ^ ^
p dx p 8x

g P — P~d ^
f p dx

(XIX. 17)

(XIX. 18)

which corresponds to a strictly equivalent-barotropic field where the boundary surface
lies at a depth —d and the current layer has a density p while the motionless layer
beneath has a density p-^. Since in the transverse section through the Gulf Stream
dpjcx is very small, the effect of the first term in (XIX. 17) will predominate and the
actual distribution will approximate closely that of an equivalent-barotropic model.

A model of this type has been worked out by Charney (1955). Earlier investigations
by Stommel (1953) and Charney (1955) have clarified the question how a boundary
current possibly might be influenced by a consideration of the inertial terms in the
total equations of motion and by stratification of the water masses.

This model assumes the j'-axis along the edge of the continental shelf with the :v-axis
at right angles ; with a slight approximation it can be assumed that the j-axis points
northward and the .v-axis eastward. Ignoring the unimportant kinematic effects of the
earth's curvature it follows from the theorem of the Constance of potential vorticity
(p. 336) that for a steady current in a water layer h above a motionless lower layer

+ v

t+f

(XIX. 19)

Introducing from the equation of continuity the volume transport stream function i/<
which is defined by

hv = ^
ox

(XIX.20)

allows (XIX. 19) to be rewritten in the form

[dx \h oxj ^ dy \h dyj

+ f

/-(</.)

(XIX.21)

where F is a function of ip to be determined. A second equation relating iJj and h is
the Bernoulli-equation. This gives

1 r/1 dipy _ /I d,p\

2 [[hd^j ^ [hdj'j

g*h = G(0),

(XIX.22)

where ^* = (ph-- p)l Ph-S and pn is the density of the lower motionless layer and p is
the density of the upper moving layer. G(^) is another function of i/*, which has to be
determined.

628

The Tropospheric Circulation

Taking into account the magnitude of the different terms in the equations (XIX.21)
and (XIX.22) these can be written simply as

1 (dv

+ g*h + G(<A),

(XIX.23)
(XIX.24)

where v is given by the second equation of (XIX. 20). The determination of the functions
F and G is laborious and requires the use of the outer (seaward) boundary condition.
Denoting quantities at this boundary by a bar, it follows from (XIX.23) and (XIX.24),
since at the boundary {x = oo) both v and dvjdp are zero, that

flh^Fi^jj) and g*h = G{^P).

(XIX.25)

For X — CO, i/j and h are functions of j^ and also fi is a function of ip, that is, F and G
are then also functions of ifj and y.

Since F and G are in principle to determinate for jc = oo they must also be deter-
minable at every point in the interior region connected with the outer boundary by a
stream line. It is therefore possible to determine F(4)) and G(ifj) at all interior points.
The function ifj is taken as a parabolic function of y which is made plausible by the
observations at the eastern edge of the Gulf Stream.

^ = ^0- y(y - y'of-

With sufficient accuracy /can be taken as a linear function of >•

f=fo + Ky-yo)

which gives finally after some calculation

and

which is valid for all values of y and ip.

The equation (XIX.23) and (XIX.24) then give the final equation

8i/j f I /dG
dh g*h g* \ bijj

Its solution, subject to the boundary conditions h = h{y) and </<

/j2 = /;2 + 1/(0 _ 0).

G'{^)=g*

/?2 -

-!■-•■

- ^)"2

1/2

(XIX.26)

(XIX.27)

(XIX.28)

- F

0( >') is

(XIX.29)

(XIX.30)

The velocity v is obtained as a function of ijj and y from the equation (XIX.24) and the
values of X corresponding to 0 and y are given by the equation

dx =

(XIX.31)

The Tropospheric Circulation 629

which, only requires a numerical quadrature; the boundary condition here is (/» = 0
at ;c = 0.

The application of this theory put forward by Chamey starts with the deter-
mination of the two constants in equation (XIX. 26). Taking 0 = 0 at the coast, then i/<
is the volume transport of the current. The zero point for y is midway between the
Florida Strait and Cape Hatteras ( y = y^), that is, 700 km from both sides.
The calculated geostrophic transport in the Florida Strait is approximately
30 X 10^ m^ sec~^ and the increase from here to Cape Hatteras is approximately
50 X 10« m^ sec-i.

Hence ^o = 80 x 10^ m^ sec~^ and y has the value 2-55 x 10"^ msec~^ Further-
more, in (XIX.27), /o = 0-84 x 10"^ sec-^ and /3 = 1-8 X lO-^^ m-^ sec-^.

If we postulate that h = 0 when x — 0, y = yo and tp = 0, then substituting these
values in equation (XIX. 30) gives

^„ = /^0„y' =820m (XIX.32)

which compares well with the observed mean value of 900 m given by Iselin (1936).

The results of the integrations are shown in Fig. 296. This gives in perspective the
calculated position of the boundary surface h by contours of h (full lines) at 100 m
intervals and on this surface the stream lines (broken lines) of the volume transport
for each 10 million m^ sec~^. On top are given calculated velocity profiles for several
cross-sections through the Gulf Stream. Comparison of the position of the internal
boundary surface with the observed mean depth of the 10° C isotherm, which gives
approximately the lower limit of the Gulf Stream, shows that they are in excellent
agreement. The characteristic way in which the current swings away from the coast in
the northern part of the region considered can also be seen. This takes place away
from any projection of the coast line and is found both in the Gulf Stream and in the
Kuroshio. The current profile shows towards higher latitudes an increasing concentra-
tion of the current energy towards the left-hand edge (westward intensification). The
velocities along the left-hand edge are probably too high in the north but would be
reasonable since boundary friction was neglected.

The theory takes a simple form if a quasi-geostrophic approximation is made, that is, when both
M and V are assumed to be geostrophic and when h varies linearly with y, then

h = ho + H-iy - yo)-
With the condition A = //, at x = 0 (at the coast) the solution is

h = 7i( >•) -(h- h,)e-x'x. (XIX.33)

The width of the current is given approximately by

Since at the right-hand edge the lateral velocity at the outer boundary is | i7 1 = (g*lf)H; one obtains

A = V(mIP). (XIX.34)

It is apparent that the Gulf Stream is a phenomenon that depends essentially on the variation of the
Coriolis parameter with latitude. Observed values of u and j3 give a value for A of about 50 km.
V decreases laterally to a quarter at a distance of about 70 km which is in accordance with the down-
slope to the right shown in Fig. 294. The geostrophic approximation predicts roughly the character
of the current but does not predict all the details.

630

The Tropospheric Circulation

The Tropospheric Circulation 631

The theory of the Gulf Stream and similar boundary currents requires further
development. The double-layered model must be replaced by one with continuously
stratified water and the effects of friction in both vertical and horizontal directions
must be taken into account. Lateral friction against the coast should give a reduction
in the velocity of the current at the left-hand side as is shown by observations.

The boundary current theory attributes the ocean boundary currents of the general
oceanic circulation, in so far as they have the character of a free jet, to the effects of
pressure and inertia and to the variation of the Coriolis parameter with latitude. It
has been pointed out above (p. 580) that the Sverdrup solution starting from an
eastern continental boundary and working westwards is unable to satisfy the boundary
conditions at the west coast of the ocean. Only by including the effects of a strong
lateral friction (mixing) Stommel and Munk have been able to satisfy the boundary
conditions at a western boundary and to give a general theory of a wind-driven
ocean circulation. However, along the eastern side of a continent (western side of
oceans) the currents apparently do not correspond to this theory. They are narrower
and more intense than would be expected from the general theory. The Charney
theory gives the explanation for this and yields in this way a western continuation to
the Sverdrup solution, without the addition of strong frictional effects but taking
into account the effects of inertial terms and the variation of the Coriolis parameter
with latitude. The density stratification of the water and the lateral inflow into a
meridionally directed jet current have been found to be of particular importance in
the formation of these boundary currents. These provide the connection with the
western transport of the zonal wind currents of lower latitudes.

{d) Further Aspects of the Dynamics of the Gulf Stream

Associated with the questions raised on p. 617 another one stands out concerning
the total current energy in a relatively thin top layer. This energy concentration in a
narrow current band occurring in the very upper layers persists for more than 2000 km,
from Cape Hatteras to the region east of the Newfoundland Banks while beneath this
top layer the velocities remain small. This remarkable phenomenon is probably
explicable by an association between momentum losses in the lower portion of the
current and the upper energy concentration. It should be stressed that the zonal width
and the high speed of the upper Gulf Stream layers rather definitely exclude an inter-
pretation of the current in this part of the Atlantic as the result of momentum added
locally by the prevailing winds. Rossby (1951) has attempted to find out what kind of
verticaly velocity profile would be formed in an immiscible stratified current subject
to momentum losses through contact with the underlying surface or at lateral boun-
daries. It would be of particular value to know the nature of the special velocity
profile corresponding to a minimum value of the momentum transfer in unit time
across a vertical plane normal to the current axis. It is reasonable to assume that
this profile represents a limiting state which would be gradually approached by any
stratified current subject to momentum losses but unable to escape to the sides.

In a straight aparallel current of this type in which the water is considered to be
incompressible and the density varies with depth, the momentum transfer across a
vertical strip normal to the current axis is given by

632 The Tropospheric Circulation

MT = r (pm2 + p) dz, (XIX.35)

where z is counted upward from the bottom and where p is the water hydrostatic
pressure. Assuming that the mass transport in every infinitesimal isopycnic layer
remains constant during the variation process, then

puz da = pUq Zq da = v{a) da, (XIX. 36)

where the subscript 0 indicates initial conditions. Here a is a new independent variable
which determines the vertical density distribution and i = dzjda. With these quantities
(XIX.35) gives

MT = f /-^ + pz\ da. (XIX. 37)

With the fundamental hydrostatic equation

one obtains finally

f = - gP^ (XIX.38)

MT^ - \ i^ +-\ da. (XIX.39)

The variation problem is the determination of the particular function p of a which
reduces MT to a minimum value for the given distribution of v with a. The variation of
p vanishes at the sea surface and it can be assumed that it also vanishes at great
depths. Under these circumstances the minimum value of MT is then given by

8{MT) =[ \(^-^-^~-]8p-^ 8p] da = 0. (XIX.40)

This is true for arbitrary values of 8p provided the function p satisfies Euler's equation

gp da

P^ gp.

0 (XIX.41)

which on substitution reduces to

du^ = p da, (XIX.42)

where a is the specific volume.

To determine the final velocity distribution from the initial mass transport distri-
bution it is necessary to combine (XIX.42) with (XIX. 36) or

pu dz = v{a) da. (XIX.43)

Rossby has discussed several models with special density distributions according to
this principle; only those more or less directly concerned with the Gulf Stream will be
considered here.

For a uniformly stratified current with speed Uq and depth D^ that is flowing on
top of a homogeneous bottom layer of density p^, in which the volume transport is
zero and that is allowed to readjust itself to a minimum momentum transfer current

The Tropospheric Circulation 633

profile, a determination of the density and velocity distribution in the final state can
be made by taking

P = p,(l + Iko) and p, = p,{\ + 2k), (XIX.44)

where Ps is the surface density and p,, is the deep water density. For a uniform initial
stratification (subscript 0) it follows that

With the continuity requirement, the basic equation gives as a good approximation

Further when o^^j^ — 1 one obtains

D I 3«2 \i/3 „ 3 m\

It follows that the current must become shallower and the bottom layer will increase
in thickness whenever Wq falls below the critical value, Mo.crit defined by

Wo < "o.orit - J^^\ (XIX.48)

The end of the adjustment process can be illustrated by means of a numerical example.
Initially the upper moving layer extends down to 600 m (Z)o = 600) and
Wq = 0-75 m sec"^ In the Gulf Stream region an adequate value of the total range
in CT< is 4-5 so that to a close approximation Ik = 4-5.

Thus D results to 300 m and for u^ one obtains 2-25 m sec~^ Figure 297 shows a
graphical representation of this case.

It is clear that the dimensionless quantity F defined by

P - IT \i X n (XIX.49)

{(pb - Ps)/pb}gDo

has the form of a Froude number in which the gravitational acceleration is reduced in
proportion to the total percentage density range of the fluid. It can be seen that this
new number determines the nature of the baroclinic movements of a current subject to
momentum losses due to frictional influence. If the "internal Froude number" is less
than a certain critical value (in the above case ^) the current will be concentrated in the
lighter top layers.

Apparently, oceanic currents usually have subcritical values of F. They then have a
tendency to develop a strong shearing motion with increasing velocity and increasing
stability near the sea surface and decreasing velocity and stability lower down.

In the Straits of Florida and in the Gulf Stream region as far as Cape Hatteras the
range in CT( is smaller than it is further downstream and there is no homogeneous deep
water to facilitate a separation of the current from the bottom. After the current leaves
Cape Hatteras, however, the momentum it gains due to direct action of the wind on the
narrow strip exposed at the atmosphere is presumably incapable of balancing the losses

634

The Tropospheric Circulation

which result from interaction with the deeper water masses or are due to lateral mixing.
The current thus tends to become more and more superficial ; this process maintains
the high surface velocities.

The cause for the horizontal meander-like oscillations of the narrow current band
of the Gulf Stream after leaving the continental shelf is not entirely clear. These
meanders occasionally become unstable and then complete cut-off vortices are formed ;

lOOm

200 m-

300 m

400m

500m

0 U

600m

mps

1-0 o-

Fig. 297. Transformation of a uniform current with a constant vertical density gradient into
a flow characterized by a minimum value of the momentum transfer. The initial uniform
velocity distribution is given by the heavy broken line, the final velocity distribution by the
heavy full line. The density distributions before and after adjustment are given by lines
marked by a (initial) and a (fmai)- Note that the depth of the final current is one half of the
initial depth. The total percentage density range has the value 00045.

this has been discussed already on p. 616. Recent investigations on the vertical strati-
fication in the Gulf Stream (Arx, Bumpus and Richardson, 1954) using stations with
little distance from each other have shown that the narrow current band has a filamen-
tary structure. It is composed of thin layers of high velocity alternating with layers of
lower velocity. This extraordinary stratification is possibly connected with gliding
processes imposed by external circumstances on the individual water layers of the
Gulf Stream and can be assumed to be a consequence of turbulence processes, which
are imposed from outside.

The meandering of the narrow current band of the Gulf Stream appears to be a
common phenomenon. These meanders show v/avelengths of about 200 km and their
speed of propagation is about 1 1 nautical miles a day, which is about a tenth of the
speed of the current itself. Stommel (1953) has given a simple meander theory for a

The Tropospheric Circulation 635

wide current in a stratified ocean in which he showed that the stability of the waves
depends on whether

C/2 > g:^D. (XIX. 50)

Here f/is the velocity of the basic current, D is the thickness of the upper moving layer
and J p is the density difference between the lower, homogeneous and motionless layer
and the homogeneous upper layer. The upper inequality sign results only in stable
waves and the lower one only in unstable waves. For W = g(Aplp)D there is a single
"just unstable" wave, the wave-number of which is given by k =f/{U\/2). This
wave always remains stationary.

Choosing a surface layer 200 m thick moving at 200 cm sec^^ and having a density
ratio zJp//3 = 2 x 10~^ the wavelength of the "-ust unstable" perturbation is 180 km.
All other wavelengths are stable and do not grow. It is remarkable that this wave-
length corresponds closely to that observed. Some objections can be raised against
the application of the Stommel perturbation theory to the meanders actually observed
in the Gulf Stream and it would be desirable to test the Stommel model somewhat
more closely and to specialize some of his assumptions.

In order to handle the problem of the meandering behaviour of the Gulf Stream in
a more comprehensive way, the problem may be looked upon as intimately connected
with the way in which the stability of a narrow geostropliic current is changed when this
flow is subjected to external perturbations. In a deeply penetrating way the latter
question has been dealt with by van Mieghem (1951) for atmospheric currents. He
assumed a straight geostrophic flow in hydrodynamic equilibrium in any direction on
the rotating earth allowing for horizontal (transversal) and vertical wind shear. On this
current he imposed a disturbance acting in lateral (transverse) as well as vertical
direction and attempted to find the conditions under which the disturbance decreased
in time (stable state) or increased in time (unstable state). In the stable case the chance
disturbances vanish with time; in the unstable case they grow into meanders and may
even degenerate into independent vortices. If the positive x-axis is chosen in eastward
direction, the >'-axis normal to it (to the north) and the r-axis positive towards the
zenith and if the geostrophic current flows along the j'-axis (w^ = 0, iiy ^ u(x,z),
u^ = 0), then the equilibrium values of the pressure P = P(x,z) and the specific
volume a = a(x,z) are only functions of jc and z and the equation of motion as well as
the quasihydrostatic equation leads to the Margules equilibrium condition of the
geostrophic current :

cPca_cPca^^^ (XIX.51)

ox cz dz ex

where oj^ and a»y are the horizontal and vertical components of earth rotation vector
(coj. = ojy = oj cos </) and cu^ = cm sin (/>) and N is the number of solenoids in the cross-
section {x,z) (baroclinicity). For a small fluid particle in the interior of the water mass
which is at the co-ordinate origin at time t^ and at that instant is subject to a transverse
impulse, its velocity components relative to the earth at the same instant will then be

V, = u-\- r„ Vy = Vy, V, = V,. '. (XIX.52)

(^z

2<^x

636

The Tropospheric Circulation

Assuming that the specific volume a^ of the disturbed particles is conserved, then the
equations of motion for the displaced particles will take the form :

where

+ 2a)yV^ = ijjx

dv^
dt

— 2cOyl\ = ^y

4'x = -

- a^^x - a^^z.

•A. = -

- a,^x - a-^^z.

}

(XIX.53)

(XIX.54)

X and z are the displacements of the small particles in the x- and 2-directions and
may be positive or negative. The coefficients a^x, ^xz and a^^ are given by

\ dxf dx 8x

--/(/♦-^l-^.

= +/* /

(^•-9

dP8a

dx dz

dP8a

Tzd^

(XIX.55)

with axz= ^zx and/* = 2a; cos ^.

It can then be shown that at a point in a geostrophic current at which there acts a
transverse disturbance, conditions will be stable, neutral or unstable according to
whether the quadratic form (Kleinschmidt) :

x2 + 2a,

+ a.

^0.

(XIX. 56)

The sign of Q is determined firstly by that of the discriminant

a ^ a^, — arr. a

'XX "xz

(XIX.57)

and secondly by the sign of one of the coefficients of the quadratic terms in Q (for
instance a^^). The condition (XIX. 56) thus becomes

a-0 or

idu _ daldx ^\ , ^>Q
\dx daldz' 8z) ■' < '

(XIX.58)

The last equation can be re-written with the help of (XIX.55) and by neglecting terms
of lower order one obtains

4:^1 /(/+ir"-

(Jadz

(XIX.59)

The expression - ^

p dz

is the static stability (z-positive upwards; p. 196) and

f{f-\- dujdx) is the expression for the inertial stability. The equation (XIX.59) gives
a hydrodynamic measure in as far as the geostrophic equilibrium in the current under
consideration is hydrodynamically stable or unstable when subject to external impulses
acting normal to the direction of the flow (in transverse or vertical direction).

The Tropospheric Circulation

637

The application of these equiHbrium conditions to the Gulf Stream requires an esti-
mate of the order of magnitude of the individual terms. These can be obtained
approximately from the "Atlantis" sections for concentrated boundaries of the current
and one obtains the following values given in the [cm g sec] -system:

10-

du
ex

da
dx '

gda

a dz

10-

10-^ to 10-8

10-=

/•

(f- a

. 10-*
. 10-1*
. 10-8

Introducing these values in equation (XIX. 59) shows that in the Gulf Stream, in spite
of always secured static stability and in spite of the almost always secured inertia
instability, hydrodynamic instability may still occur provided the vertical shear in
the flow reaches excessive values.

This can be illustrated by an example taken from the "Atlantis" section shown in
Fig. 294. (Chesapeake Bay-Bermuda, April 1932). Along the left-hand side of the
Gulf Stream in the region of largest vertical and horizontal shear (depth 220 m) one
obtains

du cu

— ==0-47 X 10-2sec-i; ^

cz ox

0-33 X lO-^sec-i and

0.33 X 10-«.

With these values and with/ = 0-85 x 10-*

[fj^^ =0•16xl0-l^

while

' CxJ a CZ

The current and density stratification is thus, of course, hydrodynamically stable as
could be expected since at this part of the Gulf Stream the current shows no tendency
to meander. Hydrodynamic instability would only occur if the vertical shear in the
flow would reach values four times larger. Further to the north, in the section between
Cape Hatteras and the Newfoundland Banks, conditions might be diff'erent and may
readily be so that the current system becomes hydrodynamically unstable; these small
horizontal wave formations will soon grow into large meanders and finally lead to the
formation of vortices. Strong vertical current shear and low static stability are required
for this. It can be understood that a strong acceleration of the flow in the top layers of
the Gulf Stream caused by the direct action of a strong westerly wind acting on the
sea surface will provide the necessary vertical current shear to give rise to hydro-
dynamic instability in the current system and to lead to the formation of meanders.

Haurwitz and Panofsky (1950) in a study of the stability and meandering be-
haviour of the Gulf Stream have attempted to show that especially favourable con-
ditions for the development of unstable waves occur when the Gulf Stream is not too

638 The Tropospheric Circulation

close to the continental shelf. The tendency towards a formation of meanders appears
only after the Gulf Stream leaves the continental shelf, but probably there are other
factors that will decide about the development of meandering motion than the distance
from the continental shelf.

As yet no fully satisfactory explanation has been given for the observed split of the
Gulf Stream into a number of branches. Hansen (1952) has demonstrated that under
certain conditions a northwards flowing current while turning towards the east can
break up into several branches; but his solution is of more formal character and no
actual reasons can be offered for this phenomenon.

{e) The Kuroshio

The three-dimensional structure and the dynamics of this current have been
investigated by Uda (1930), Sigematsu (1933) and Kisindo (1934) on the basis of
series observations made by the hydrological department of the Japanese Marine and
the Imperial Fisheries Experimental Station in Tokyo (since 1925) and also by the
oceanographic survey vessel "Mansyu". A number of transverse profiles have been
prepared and critically worked with by Wust (1936a) in a comparative study of the
Kuroshio and the Gulf Stream and further valuable work has been performed by
KOENUMA (1939). Wiist has dealt with a cross-section at right angles to the chain of
islands, the Ryu-kyu, from 27° to 29° N., just before the Tsusima current splits into
branches and with another cross-section farther north (little to the south of Shiono
at Misaki, the south cape of the projecting Kii peninsular at about 30° to 34° N.).
See Fig. 261 for the position of these sections.

The inclination of the isolines of the oceanogi-aphic factors forced by the water
movement appears clearly in all cross-sections through this strong current. A com-
parison with conditions in the Gulf Stream shows that there is an almost identical
thermal structure but considerable differences occur in the salinity distribution; the
Kuroshio has a low salinity 34-32 to 34-98%o and a very weak vertical salinity stratifi-
cation, while the Gulf Stream possesses considerably higher salinity (34-97-36-65%o)
and a pronounced stratification. The Kuroshio region also shows an intermediate
salinity minimum at 500-800 m depth resulting from an intrusion of the weakly
saline sub- Arctic intermediate water flowing in from the north (p. 172).

Figures 298 and 299 show the temperature and salinity distributions in the Ryu-kyu
section (Feb. 1927) and in the Shiono-Misaki section (Jan. 1927). Disregarding the
top layers, the sections for the summer months show entirely similar conditions.
These sections have also certain similarities with those through the Gulf Stream (see
Figs. 282, 283).

The Ryu-kyu section corresponds closely to that through the Florida Strait, the
Shiono-Misaka section to the Chesapeake Bay transverse section. It is also apparent
from these sections that the Kuroshio is throughout the entire vertical extent a weakly
saline current as compared with the Florida Current; the highly saline core layer can
again be explained as a distant effect of the tropospheric circulation of the subtropics
and tropics. The velocity distribution calculated from the mass field of the Ryu-kyu
winter section shows maximum intensities of 61 cm/sec below the sea surface at
150 m depth. In summer highest values of about 90 cm/sec occur at the sea surface.
The weakening and downward displacement of the current maximum in winter is in

The Tropospheric Circulation

639

439 438-

4-_ - ^>22^2j^ f" 11 /\ m — :ir~~"

Fig. 298. Cross-sections of temperature through the Kurochio (R, Riu-Kiu section at 28^

to 29° N., "Mansyu" stations; S, Shiono-Misaki section at 34" to 30" N., "Mansyu"

stations, January 1927) (according to Wiist).

Station

KDOO

Fig. 299. Cross sections of salinity through the Kuroshio, section S (Shiono-Misaki) (see

remarks below the Fig. 298).

640 The Tropospheric Circulation

correspondence to the piling-up effect ("Aufstau-Effekt") of the winterly north-west
monsoon. These values are in good agreement with direct current measurements at a
station in the current core. The total amount of water transported through this section
amounts to 21 million m^/sec in winter and about 23 million m^/sec in summer.
The Kuroshio and the Florida Current thus carry about the same amount of water.

The Shiono-Misaki section has been evaluated both by Wiist and by Koenuma.
WUst thereby placed the reference-level at the upper limit of the weakly saline inter-
mediate water, at about the depth of the 10° isotherm; Koenuma on the other hand,
bases his calculations on velocities of 16 cm/sec of the intermediate water observed
in coastal areas moving there to the north-east and for larger distances from the coast
he assumed that the intermediate water was transported to the south-west at 5 cm/sec.
The two vertical velocity profiles independently found by both methods thus do not
agree. The velocity distribution obtained by Koenuma is in good agreement with
actual current measurements while the values obtained by WUst are somewhat too
low. The Kuroshio here keeps closely to the coast with velocities of 160-180 cm/sec
and extends seawards for 140 km. As is true for the Gulf Stream, there is a counter
current to observe towards the south-west on the right-hand side with maximum
velocities of up to 20 cm/sec. Here also a downstream increase in the water transport
can be noticed, but the counter current on its right-hand side with its higher velocities
compensates the outflow towards the east to a considerable extent. There is so far no
proof whether there are any seasonal changes in the amount of water transported
(see also, in this connection the works of Ichiva, 1953-54).

The Kuroshio does not show such pronounced characteristic properties as to be
termed without more ado as a free jet current in the sense of the Rossby theory. It
lacks especially the jet-like outflow from a narrow sea strait; it is formed instead by the
gradual deflection of the stream lines out from the North Equatorial Current and only
at a later stage forces its way into the relatively narrow channel-like region between the
shelf and the submarine ridge of the Ryu-kyu Islands. By the further weakening due
to the separation of the Tsusima branch its quasi-jet character is entirely lost.

The continuation of the Kuroshio out into the Pacific from about 35° N. onwards
(see p. 570), according to vertical sections (Uda, 1935), possesses the character of a
relatively narrow current which, however, like the Gulf Stream in the central parts of
the Atlantic, has a tendency to break up into single-current branches intermittently
separated by vortices and counter currents. The one branch turning north from the
Kuroshio meets the cold water masses of the Oyashio, and there in dynamic respect
similar conditions occur as are present when the Gulf Stream meets the Labrador
Current off the Newfoundland Banks.

Table 153 finally presents a survey about mean water, heat and salt transports
according to Wiist for the Gulf Stream and the Kuroshio. About 22 times as much
water passes through the Kuroshio section and even about 33 times through the
Gulf Stream as is carried by the water transports of all the rivers and glaciers on the
earth (run-off from the continents on the average about 1-2 million m^sec). Even
more spectacular are the enormous amounts of salt carried through these cross-
sections, corresponding roughly to loads of 79,000 and 121,000 rail-road goods
wagons respectively, each of which takes 10 tons. The question thus arises, why the
climatic effect of the Kuroshio on the eastern Pacific and on the neighbouring continent

The Tropospheric Circulation

641

Table 153. Mean water, heat and salt transports of the Gulf Stream and of the
Kuroshio between 27° N and 37° N.

Water amount 10* m^/sec .
Heat amount 10^° kg cal/sec
Salt amount 10® tons/sec

Gulf Stream

(Florida and

Cheapspeake

section)

Kuroshio

(Ryu Kyu
section)

Ratio between
(Kuroshio : Gulfstream)

1 : 1-46
1 : 1-44
1 : 1-54

is so much weaker than the corresponding effect of the Gulf Stream on the Eastern
Atlantic and on Europe, although the heat transport is not appreciably less. This
difference must be governed by topographical conditions (Dall, 1881, Koppen, 1911).
After leaving the Japanese coast at 35° N. until it diverges northwards and south-
wards on the eastern side of the ocean the Kuroshio water travels about 8000 km,
while the Gulf Stream water after leaving the American west coast travels only about
5000 km. Beneath the Kuroshio waters there is weakly saline, cold sub-Antarctic
water, but beneath the Gulf Stream the water is warmer and more saline and con-
tinuously renewed by the outflow of the highly saline European Mediterranean
waters (see p. 529, Fig. 245). The Gulf Stream water is thus protected from consider-
able heat and salinity losses downwards. The greater efficiency of the Gulf Stream must
be attributed to the much longer conservation of its properties over the considerably
shorter distance it travels and to the favourable conformation of the European coasts.

(/) The Agulhas Current

This current is due to the outflow of the water piled up by the South Equatorial
Current of the Indian Ocean along the coast of South Africa and Madagascar and as
such is a typical gradient current. A detailed dynamic evaluation of the observational
data available from the different expeditions has been carried out by Dietrich (1935).
For the surface currents see p. 567 ; for the structure and dynamic of it see p. 470,
Figs. 205-7. As subtropical and Antarctic water masses are situated side by side the
three-dimensional mass distribution is a rather complex one. Everywhere along the
African continental slope as far as the latitude of Capetown there is a steep rise of
heavier water (cold, but weakly saline) towards the coast. Towards the Agulhas
Bank the slope is flattened out and on the shelf itself is occasionally superimposed by
lighter water brought in from the south and south-east by the wind. To the south of
this heavy water mass there is found a relatively lighter (warmer, but more saline)
water mass of subtropical origin in a trough-like fashion bordering on the denser
sub- Antarctic water which moves eastwards in the south. Figure 205 shows the distri-
bution of the specific volume anomaly in a cross-section oriented from Capetown in
south-westerly direction. All cross-sections through the current are of similar nature
as this one. The depth of the trough-like confined mass of the lighter water body
(corresponding to the schematic picture of Fig. 204) is about 1000 m. Underneath
this, weakly saline sub-Antarctic intermediate water spreads out everywhere, in which
the salinity minimum weakly follows the trough-form and the rise towards the coast.

642 The Tropospheric Circulation

Since the sub-Antarctic water forms an almost zonal boundary to the lighter water
mass in the south, the trough of lighter water is narrowed towards west by the African
continent, until it finally takes almost a wedge-form at the southern peak of the
Agulhas Bank. In the further course this wedge then splits into three separate branches
with simultaneously occurring vortex formations; the southernmost of these intrude
into the heavier sub-Antarctic water and the northernmost intrude into the sub-
tropical water of the South Atlantic. The lighter water thereby decreases considerably
in thickness.

A dynamic interpretation of the above-mentioned section running south-west of
Capetown has been attempted in Fig. 206 ; similar scientific evaluation of the other
sections gave results in agreement with this. The nature of the current is shown more
clearly by the dynamic topography of the isobaric surfaces. Figure 300 shows the
dynamic depth anomaly for the 200 decibar-surface relative to that of the 1000 decibar-
surface; the first one can be taken as an approximation to the absolute topography
of the 200 decibar-surface. According to this the Agulhas Current at the 200 m depth
flows with intense velocities along the continental coast as far as the southern tip of
Africa. However, it thereby diminishes rapidly its mass and velocity and finally loses its
current character forming three large quasi-stationary vortices, the cores of which are
identical with the three branches of lighter water mentioned before. According to
Dietrich about three-quarters of the water masses of the Agulhas Current, transported
at the southern tip of Africa into the South Atlantic, is drawn into these vortices and
after mixing with the current of the higher latitudes returns to the Indian Ocean.

Analysis of the pressure distribution in the current interior shows it to be the resul-
tant of two components. The first is an effect of the internal pressure determined by
the mass distribution, and corresponds to the normal pressure distribution in a system
in which a lighter motionless water mass is embedded between two denser moving
water bodies. The second component corresponds to a ridge of high pressure occurring
in the boundary region between the two currents flowing in opposite direction and is
due to the piling up of water. Since the Agulhas Current in the northern part of the
current system as well as the broad oceanic West Wind Drift in its south both give
a total water transport towards left. In the boundary region between them water
accumulates giving rise to the second pressure component. In combination with the
first a total pressure distribution is generated which is characteristic for that found in
the Agulhas Current. Especially typical is the circumstance that the two adjacent
currents of opposite direction face each other with their faster moving parts. The large
lateral shearing forces thus formed give rise to large vortical movements (p. 570) in
which most of the flow energy is dissipated.

Dietrich, 1936 has given a comparative discussion about the structure and move-
ment of the Gulf Stream and of the Agulhas Current and reference is made to this
investigation here.

4. Upwelling Phenomena

A characteristic phenomenon occurring in the narrow oceanic strips off" the western
coast of the continents in middle latitudes is the observed cold coastal water, wliich due
to its influence on the atmosphere is of considerable climatological importance. Until
recently the investigation of these phenomena had to be based on surface data only.

The Tropospheric Circulation

643

o t:

"3 fc

o.
o
H

644 The Tropospheric Circulation

which was not enough to afford any insight into the inner mechanism of this phenome-
non. Some data for the area off Chile and Peru have been obtained by the last "Car-
negie" cruise (Sverdrup, 1930) and the "Meteor" expedition during the spring of
1937 made six profiles at right angles to the coast with the objective to study the upwell-
ing water phenomenon off the north-west coast of Africa (Defant, 1936a). Detailed
systematic investigations of the strong upwelling phenomena off the Californian coast
have been made since 1937 by the Scripps Institution of Oceanography (Sverdrup,
1938a, Sverdrup and Fleming, 1941). These cover the development of upwelling
phenomena in successive surveys and have provided some understanding of the
dynamics of the upwelling process. Some comments might be made here on individual
regions with upwelling. A summary for the oceanic regions off south-west Africa has
been given by Defant (1936a), see also, Bobzin, 1922). The surface temperature
conditions are given in the charts of the "Meteor" Report, vol. v. Atlas. In all months
the low temperatures occupy the total width of the shelf (about 100 nautical miles),
at the continental slope occurs the rapid rise to the higher temperatures in the west.
During every month the temperature anomaly is highest at the coast with maximum
values of — 8°C to — 10°C. The area of maximum anomaly moved in a meridional
direction during the course of the year: in the summer (January) it occupies its
southernmost position and is strongest between Table Bay and Luderitz Bay (32° S to
23° S.); in winter it moves furthest to the north (between the Luderitz Bay and Walvis
Bay, 27° to 14° S.). During the entire year the current system of the sea surface shows
a particularly characteristic one-sided divergence line which extends along the coast
from about 30° to 20° S. or even more. In the south its distance from the coast amounts
to about 160 nautical miles; in the north, however, 300 to 360 nautical miles. The region
to the east of this divergence line is the region of cold upwelling. Where the unilateral
divergence is most strongly developed, also the temperature anomaly is greatest.
The anomaly at the coast vanishes north of 20° S., where the divergence with a de-
creasing intensity turns westwards and gradually fades away. The uniform rise of the
isopycnals from west to east (towards the coast) is a particularly marked feature of the
thermo-haline structure of the upwelling region. Off the coast especially in the north
there is a well-developed transition layer, and all the isolines immediately beneath
this transition layer off the coast show a surprisingly sharp downward deflection to a
depth of 350 m. This is only explicable as an effect of piling up of water at the conti-
nental slope whereby in the depths lower than 30 or 40 m the water masses are pressed
downwards.

Similar conditions apply also to regions with cold water upwelling off the north-
west coast of Africa. From January to May especially this region can be visualized
by a tongue of cold water extending from higher latitudes southwards along the coast.
Figure 301 shows this temperature anomaly for April ; it occupies the entire area between
the Canaries and Cape Verde in which the anomaly already on the average is increased
to almost — 7°C just off the coast and for individual cases reaches values of — 10°C
or more (see Schumacher, 1933). Here also a sharp density transition layer can be
found extending along the edge of the shelf until just off the coast.

Particularly well-developed upwelling phenomena occur in the region off the
western coast of North America between about 46° N and 25° N., especially off Cali-
fornia with extreme conditions at Cape Mendocina (north of San Francisco). An

The Tropospheric Circulation

645

analysis of the thermal conditions in this oceanic region has been carried out by
Thorade (1909) and McEwen (1912, 1934). The onset of upwelling phenomena usually
occurs in March and reaches its maximum during the summer months (July to August).
The culmination coincides with the maximum frequency of the north-west winds. It is
absent during the autumn and winter although off-shore south-easterly winds are not

40° 35" 30 25° 20" 15 10" 5° 0° 5°

Fig. 301. Mean anomaly of the sea surface temperature off the north-west coast of Africa
for April (drawn from means of two degree squares of the Atlantic Ocean).

uncommon. The cold upwelHng water off the South American coast has been dealt
with by GuNTHER (1936) (see p. 571). The west coast of Austraha is not entirely free
of cold coastal water as has been shown by Schott (1933) and rising water sometimes
occurs off the north-western coast. Occasional observations of cold upwelling water
have also been made along many other coasts, for instance, off the Somali coast
during the summer months during off-shore winds and at the southern tip of Ceylon
and others.

In considering the dynamics of the phenomenon it should particularly be remem-
bered that for a current in stratified water the mass field adjusts baroclinic, so that

646 The Tropospheric Circulation

under stationary conditions the lower and cold as well as nearly always weakly saline
waters are lifted on the right-hand side of the current core in the Northern Hemisphere
and on the left-hand side in the Southern Hemisphere. If there is a parallel coast along
this special side of the current the water off the coast already for this reason alone will
be colder and will have a lower salinity than further out. This state does not represent
an upwelling phenomenon, but rather a state of long duration dependent on the nature
of the vertical water stratification and on the current strength. Most of the anomalies
appearing off the coasts are due to such a simple effect on the mass field produced by
the currents. Upwelling of cold deep water occurs only if in a wind-driven current
with a flow component parallel to the coast a water transport away from the coast
sets in. The continuity condition then requires a rising water movement at the
coast.

In a first attempt in order to explain this phenomenon Thorade, 1909 used this
theory, and later on particular interest has been devoted to the determination of the
vertical velocity profiles in the rising water (McEwen, 1912) and to the determination
of the depths in which the upwelling phenomenon starts out (Sverdrup, 1930). It
was soon found out from the thermo-haline structure in the upwelling region, that
these depths could not be large and that due to the inclination of the isothermal layers
off the coast an upward water movement of only a few hundred metres would be
sufficient to explain the observed sea surface anomaly. The formation of a one-sided
divergence line running more or less parallel to the coast is the characteristic feature of
the current field. The occurrence of rising movements at divergence lines in the case
of non-stationary discontinuity surfaces and vortices is, of course, understood
theoretically (p. 469) and water movements of this type are shown definitely by
numerous observations of the vertical and horizontal distribution of the oceano-
graphic factors (for example, equatorial cold tonges in the Atlantic and Pacific
(pp. 558 and 569); boundary regions at the oceanic polar fronts, p. 471).

In the upwelling regions off the west coasts of continents all upwelling phenomena
are of a similar type as discussed above. From the analysis of the mean oceanic state
off the coast of South West Africa Defant (1936^) has derived the schematic diagram
shown in Fig. 302 of the structure and the water movements in a cross-section at
right angles to the coast. Essentially the cross-sectional movement consists of an
elongated vortical motion around a horizontal axis which is superimposed on a
much stronger and uniform current parallel to the coast. The water beneath the
axis of the transverse vortical motion flows in the lower part of the top layer, in the
density transition layer and beneath it towards the coast and gradually rises just off
the coast. The upwelling phenomenon is very largely confined to the narrow strip
between the divergence line and the coast. It rises up to the sea surface from a depth of
only 100-200 m and as a consequence of the current field the temperature distribution,
observed in vertical direction remote from the coast, is twisted around and changes
its position into a horizontal one; so to say is projected on the horizontal sea
surface.

A necessary consequence of this circulation is the destruction of the density transi-
tion layer in the upwelling region off the coast. This is clearly shown by the "Meteor"
cross-section (1937) over the shelf off the north-west African shelf. The gradual break
down of the transition layer, which at times is also strongly developed in the area

The Tropospheric Circulation

647

Divergence

Horizonol temp, distribution, °C
■^15° 14° 13° 12° 11°
-3° -4°-5°-6°-7^jemperature anomaly,

-100

--200 Q

-300

500

400 300 200

Distance from coast in Sm

100

Fig. 302. Schematic cross-section normal to the coast of south-west Africa. Full lines,
isopycnals ; arrows, zonal and vertical velocity components (the length of the arrows can be
taken approximately as a measure of the speed) ; letters, meridional velocity components and
in special ; A^, parallel to the coast towards north ; S, parallel to the coast towards south (the
size of the letters can be taken approximately as a measure of the speed) ; wavy lines, axis of
the vertical current vortex (vertical exaggeration 1 :2300).

nearest the coast, is a consequence of internal tidal waves which gradually become
unstable as is definitely shown by the series of observations. This is thus a precondition
for the upwelling of deep water (see vol. ii, p. 581).

SvERDRUP (1938a) in the evaluation of the almost synoptic surveys made by the
Scripps Institution of Oceanography, La Jolla, from March to June 1937 along a
transverse section off and at right angles to the Califomian coast from Port San
Luis (35-2° N., 120-7° W.) has obtained good insight into the dynamics of the up-
welling processes. Figure 303 presents two topographies of the physical sea level as well
as the 100 and 200 decibar-surfaces relative to that of the 500 decibar-surface. In the
time between the two surveys typical mass displacements have occurred. The changes
in the profile occurring down to the 200 decibar-surface can only be interpreted by a
water transport away from the coast and by the piling up of the lighter surface water
near Sts. 4 and 5. These movements can be looked upon as a consequence of the winds
which blow with little variation for long periods, on the average from N. 23° W. at
about 6-7 m/sec, almost parallel to the coast. According to the Ekman-theory under
these conditions a transport directed away from the coast can be expected. This trans-
port can be derived from the change in the course of the density lines between the two
surveys. These surface waters are carried outwards and piled up about 100 km off
the coast.

From the analysis of all the fields Sverdrup has derived the mean current field
shown in Fig. 304 during the period between the surveys. The calculated maximum
transverse velocity seawards thereby amounts to 1 1 cm/sec, in good agreement with
the velocity of the wind drift. Between the coast and the water piled up further out

648

The Tropospheric Circulation

ST NO I

040

t-0-35
0

0-90

085

-oeo

0-75

u
060 -

5 100 D-BAR. OVER 500 D-BAR
<

0-55 o

200 D-BAR. OVER 500 D-BAR.

DISTANCE FROM COAST IN KM.

100

150

200

250

Fig. 303. Topography of the physical sea surface and of the isobaric surfaces (relative to the
500-decibar surface) for the oceanographic surveys. I, 25-26 March 1937, and II, 5-6 May
1937, of the profiles through the Califomian region of upwelling water (according to

Sverdrup).

STNXii

Fig. 304. Computed mean vertical circulation for both profiles I and II in the cross-section
through the Califomian region of upwelling water (according to Sverdrup). The direction
of the motions is indicated by the thick lines with feathers; the horizontal velocities are given
by the thin lines. The region indicated by -f -f -f + + shows a zone with stronger flow
parallel to the coast and directed into the picture.

The Tropospheric Circulation 649

there is a partly closed circulation down to a depth of 80 m. In the upper half of this
circulation the water flows away from the coast, in the lower half towards the coast.
Near to the coast the water rises and in the region remote from the coast it sinks
along a boundary layer. This outer boundary layer itself moves away from the coast
and as a compensation a replacement has to be made from below (from depths of not
more than 200 m). In other cases dealt with by Sverdrup conditions are somewhat
more complicated but the essential characteristics are retained.

In a study of the large amount of observational data, on the Californian upwelling
region, collected by the Scripps Institution of Oceanography in La Jolla, Defant
(1950, 1951) it has been shown that the piling up and upwelling processes are associated
with characteristic displacements of the sea surface and of the internal boundary layer
which gradually develop under wind influence and adjust with simultaneously formed
and normal to the coast occurring circulations. They finally tend towards a stationary
state. These condition can be illustrated by two opposite cases. During the first cruises
(28 February to 15 March 1949) it was found that the wind component towards the
coast predominated over the entire region with a maximum of 5 m/sec and caused
considerable piling up of water along the coast. During the second cruise (27 April to
15 May 1949), in contrast to the first case, the water was driven away from the coast
where as a consequence upwelling occurred.

Cruise 1 thus is a typical example for a water accumulation along the coast, while
cruise 2 is typical for coastal upwelling. Figure 305 shows the dynamic topography of the
ocean surface represented by lines of equal positive and negative deviation from the
basic distribution produced by the Californian Current flowing south. This basic
distribution has been obtained by elimination of the disturbances caused by tide waves
and internal waves (Defant, 1950). The two cases show completely opposite trends.
First of all it may be noticed that the channels of positive and negative deviation
(shown by the contours) are more or less parallel to the coast following the wave-like
form of the disturbance, thereby forming a marked regular pattern. In cruise 1 the
coastal strip shows a pronounced positive deviation — with maximum values at the
coast. Outside this there is a strip of negative deviation, then farther out a strip of
postitive deviation, and finally a second negative strip forms the western border of the
region. Cruise 2 gave the same pattern with the signs reversed.

In cruise 1 there is undoubtedly a piling up of water at the coast ; it was fully developed
at the beginning, but during the remainder of the cruise (about two weeks) it could be
maintained to this extent only if the tangential wind stress towards the coast exactly
balances the pressure gradient of the sloping physical sea surface. The water masses
piled up on the continental shelf are drawn from the oceanic strip just off the conti-
nental slope; there the sea level consequently lies somewhat deeper (trough-like
form). This disturbance then develops wave-like oscillations farther westwards and
generates the adjoining disorders. Exactly the same applies to cruise 2 but instead of
piling up of water a depression in water level occurs. Consequently, to these primary
disturbances the adjacent displacements in the sea level thus take place in the reversed
order.

The dynamics of the processes of upwelling and removal of water as a surface drift
requires that the rise and fall of the physical sea surface should be accompanied by a
corresponding fall and rise in the density transition layer. In these processes (close to a

650

The Tropospheric Circulation

Fig. 305. Position of the physical sea surface and of the internal thermohaline boundary
surface and the corresponding circulation cells of the upper layer during the cruises 1 and 2.
In the first case: "Anstau" at the coast (piling up of water); in the second case: upwelling
off the coast. The inclinations of both boundary surfaces are strongly exaggerated, that of
the physical sea surface by far more than that of the thermocline.

Stationary equilibrium) in a sea composed of two layers, the displacement of the physical
sea surface is always inverse to that of the internal discontinuity surface. However,
the fluctuations of the internal discontinuity surface is many times greater (inversely
proportional to the difference in density of the two water masses). Figure 306 shows a
schematic cross-section for cruises 1 and 2. The effect of the wind on the sea surface
gradually builds up to such a stage where the wind effect is exactly in balance with the
developing pressure gradients. While approaching this stage circulations have developed
mainly in the mixed layer, and must take the form shown in Fig. 306. On cruise 1 the
water accumulation at the coast causes a downward circulation here and a sinking of
the density transition layer. Upwelling occurs in the trough forming outside this
region of accumulation.

In contrast to these conditions, during cruise 2 the water is driven away from the
coast, where upweUing thus takes place and the water masses sink down in the accumu-
lation region away from the coast. These primary circulations at the coast are followed
further out by successive secondary circulations of diminishing intensity.

The Tropospheric Circulation

651

652 The Tropospheric Circulation

To the Dynamics of UpweUing

There are a number of causes for the vertical water movements in the ocean. For
continuity reasons these vertical motions are closely connected with the divergence
and convergence of the surface waters, and there is no doubt that the upwelling and
sinking of oceanic waters is primarily connected with convergence and divergence
regions occurring at the sea surface. The cause of these divergences and convergences
in most cases lies in the distribution of wind stress exerted by the prevailing wind on the
sea surface. A totally satisfying explanation of upwelling at continental coasts has not
yet been given, and is probably not possible at all since the total process is composed
of a number of substages each of which is always controlled by other factors. Coastal
upwelling is confined to a narrow strip close to the coast (less than 100 km) and must
therefore be regarded as a boundary phenomenon.

It is a known fact that winds blowing at a suitable angle to a coast will carry light
surface waters away from it and the water mass transported away must be replaced
near the coast by heavier subsurface water. Defant (1952) gave a theoretical explana-
tion on the assumption of a sea composed of two layers with different density; previous
to this a more general investigation was made by Jeffreys on the effect of a steady wind
on the surface of a homogeneous ocean near the coast. The application of a theoretical
model as simple as this showed that the stationary wave disturbances at right angles
to the coast take their origin from the piling-up region or the upwelling region
("Anstau oder Auftriebsgebiet") near the coast (see Fig. 306) and gave results in good
agreement with those obtained by observation.

A theory of the upwelling produced by a wind parallel to a coast has been given
by HiDAKA (1954) whereby the effect of the earth's rotation and the frictional forces
due to both vertical and lateral mixing have been taken into account. He deals only
with a case of a steady state. The equations of motion, together with the equation of
continuity and the boundary conditions which must be satisfied at the sea surface and
along the coast, give a rather complicated solution to the problem. Calculation of the
magnitude of the off-shore currents and the upwelling velocity for a numerical
example allows the results to be compared with values estimated correctly from obser-
vations. Figure 307 gives the solution in the form of stream lines in a vertical plane
perpendicular to the coast. Upwelling develops close to the coast and there is no
off-shore movement of the water in the upper layers of the sea directly beneath the
surface swept by the wind. The upwelling is confined to the strip until 0-5Z)„ from the
coast and the sinking process occurs outside the wind zone. If the vertical mixing co-
efficient ^4^, is chosen with a value of about 1000 then the vertical Ekman frictional
depth Z)^, will be 162 m at 30° N. For a horizontal mixing coefficient A^ = 10^ the
horizontal frictional depth will be about 162 km. Estimation gives the width of the
coastal upwelling region as ID^ = 339 km. From this the average velocity between
the surface and the layer 0-2Z),, can be calculated as 3-35 cm/sec (off-shore the maxi-
mum upwelling is 2-7 m/day upward or approximately 80 m/month). Sverdrup
(1938) obtained a similar large value for the upwelling velocity off southern California.
The depth at which the upwelled water originates is about 200 m which is also in fair
agreement with observed values off the southern Californian coast. Hidaka has also
investigated the cases arising when the wind is at certain angles to the coast. If the
wind is at right angles to the coast, then the induced circulation has a rather complicated

The Tropospheric Circulation

653

Fig. 307. Upwelling as induced by a wind parallel to the coast illustrated by the stream lines

in the vertical plane perpendicular to the coast. In the numerical example D^= 162 m and D^

= 162 km; the width of the coastal wind belt is about 340 km.

Structure with two vortices in the upper layers, one of which is situated close to
the coast and the other near the outer boundary of the wind belt. The upwelling due to
a longshore wind (Fig. 307) is far more effective in lowering the temperature of the
coastal region than that induced by an off-shore wind, since the former one brings a
larger amount of colder water to the surface from deeper levels than the latter. This
theory put forward by Hidaka deals only with the stationary case; no attention is
paid to the water stratification which as shown by observations plays a decisive role
for the processes involved before a steady state is reached.

The process of upwelling is shown by observations to be variable with time. If
the duration of the wind is as short as a few hours, the off-shore component of surface
water transport will not be very large since drift currents will not fully develop. If the
winds are more or less steady for several hours up to as much as a day, the drift
currents may develop but they will not be followed by considerable upwelling because
of oscillations of the thermoline. However, the process will be different if the wind
continues for several days up to a week. If the wind continues for a longer time-
interval than about a week, the surface currents will reach a steady state with an inter-
mediate stage for a wind lasting a few days up to a week during which the geostrophic
equilibrium is approached. This latter section of the process has been dealt with
theoretically by Yoshida (1955) using the conditions in Californian waters as a guide.
In his model the .v-axis is directed eastwards, the >'-axis directed northwards and repre-
sents the coast line. The r-axis is chosen positive downwards with z = 0 being placed
along a mean sea level. The conditions were taken as constant in a north-south

654 The Tropospheric Circulation

direction. In addition at this stage only small-scale processes, i.e., processes extending
over a period of several days to a week and over a distance of up to 10 km, were
considered of interest. The equations of motion are then

-fv = - I (XIX.60)

dv
8i

8 / 8v\ r„

A is the eddy viscosity, Ty is the northward component of wind stress, T and H is the
average thickness of the mixed layer. The corresponding vorticity and divergence
equations are

dt, fwn curl^ T

dt H H

(XIX.62)

/^ - g, (XIX.63)

where Wf, is the vertical velocity at z = /? (depth of the thermoline). The equation of
continuity and a condition for the quasi-isostatic adjustment with g* = g(Ap/p) give

1 8p
w,^-,^. (XIX.64)

The mutual adjustment between the pressure and the current seems to be completed
within a period of one to two days, so that the above equation is reasonable for up to
about a week after this first stage of adjustment is over. From the equations (xix.62-
64) is obtained

where k =fl\^{g*H). The boundary condition along the coast {u — 0) will require

(

8w\ _A:2

with the condition w = 0 when x = -co the solution of (XIX. 65) will be
k^

.^j

fh Cx ro

Ty e^a^-f)^! + Ty e-^(^-f) di + e^^ Ty e^^ d^
0 J — CO J — CO

(XIX.66)

It can be shown that

__ 1 ej;

^^ ~ y8x

for values \kx\ > 1 and along the coastline we have

Wo = y [" Ty e^^ dx . (XIX.67)

A uniform northerly wind over off-shore water will give rise to a coastal upwelling

given by

-« = ^, • (XIX.68)

The Tropospheric Circulation 655

The upwelling velocity will be proportional to the intensity of the northerly wind but
is not directly dependent on the latitude. When g* = g{Apjp) — 2-5, i/ = 40 m =
4 X 10^ cm and Ty,Q = —0-5 then

H'a-^o = — 5 X 10"^ cm sec ~^.

In five days this upwelling will give an upward displacement of the thermoline of
22 m. This upward movement of the thermocline off the coast will continue until an
equilibrium is reached in about a week and according to observations seems then to
be maintained for about one or two months. The region where this coastal upwelling
occurs is confined almost entirely within a narrow strip close to the coast. With the
numerical values introduced above, k will result to '^0-7 X 10~^ cm~^; at a distance
of 40 km, w will be reduced to 6% of that at the coast and to only 3% of the coastal
H-value at 50 km. The process is practically limited to a distance of 40-50 km from the
coast. The effective width of coastal upwelling is given by a characteristic length

Yoshida also investigated the changes in surface conditions which were derived
from the above model of a transient state of upwelhng. He found that the variations in
surface characteristics were largely confined within the narrow coastal regions. The
coastal upwelling is associated with considerable changes in surface conditions within
the coastal waters of width L, while upwelling or sinking outside this strip will not give
rise to such significant changes during a period of only a week or two. In the succeeding
stage of the upwelling process, in which now the isostatic adjustment can be con-
sidered a complete one, the laterial mixing process in the inshore regions stands out as
the most important factor. The dynamic equations are now

- A- = - I , (XIX.69)

ft^ = ^ + A,-^„ (XIX.70)

where A,, is the coefficient of lateral mixing. The upward movement of the thermoline,
due to the ascending motions, will produce a sharp horizontal density gradient and
when conditions are variable in an oscillatory way, as is usually the case, internal
waves will originate and cause intense mixing across the thermocline. The equation
for the conservation of mass will now become

or, approximately

w ^ - An

dx"

The boundary condition at the coast gives Tq = 0 so that finally

^^ = -g-dx' ^^^^-^^^

656 The Tropospheric Circulation

The equation for w will become the same as in the earlier state and the vertical
velocity distribution will therefore remain unchanged throughout the whole period of
upwelling process as long as the wind is kept steady. During this period the ascending
water movement will be subject to mixing with the surrounding waters and the thermo-
line will not be raised to any large extent. From equation (XIX. 71) it follows that at
this stage the vorticity in the surface layer will be proportional to the vertical velocity.
Upwelling will thus be associated with cyclonic vorticity in contrast to the initial
inshore increase in negative vorticity produced by the coastal upwelling. This approach
developed by Yoshida undoubtedly appears to give a deeper insight into the dynamics
of the upwelling process, but a more specific representation in detail of these processes
would be desirable.

5. Processes at the Polar Boundary of the Subtropical Convergence Region

The subtropical convergence regions are oceanic areas where the oceanographic
factors show large local and time variations (p. 575). They can be interpreted as con-
sequences of vortex formations between the two somewhat different types of water on
the polar and the equatorial sides of the convergence region. On the one hand, there
are intrusions of warm highly saline water from lower towards higher latitudes, and on
the other hand, intrustions of cold and weakly saline water occur in the opposite
direction. All the isolines of the oceanographic factors and the isolines of the dynamic
topography of the pressure surfaces thus show a wave-like structure. Whether all the
deviations from a smooth curved pattern are of an aperiodic nature propagated in
one direction along the boundary region between the two water types and in time
dying out, cannot be decided without a rapid succession of synoptic surveys. Since
series-observations, made in the convergence region at quite different times, can all
be combined without excluding any large number of individual values into closed
comprehensive representations ; it may be safely concluded that the disturbances are
often quasi-stationary vortical disturbances whose position and extent are probably
determined by external factors.

These wave-form disturbances are particularly well developed in the convergence
region of the South Atlantic. The topography of the physical sea level between 25°
and 50° S. (Fig. 308) shows the irregular wave-like patterns in the course of the dynamic
isobaths. This starts suddenly off the broad Patagonian-Argentinian shelf and extends
across the total width of the Atlantic to the region south of Africa. According to the
topographies of the deeper levels these wave-form disturbances reach down to con-
siderable depth but their intensity decreases rapidly with depth. They can hardly be
detected in the topography of the 1400-decibar surface. Their greatest intensity is
always found in the top layers where they must originate and therefore the reason for
their formation must be looked for here. The entire oceanic structure is shifted towards
the poles and the equator, respectively, by the interacting intrusions of different water
masses in a strip-like manner, and thereby differently stratified oceanic spaces oppose
each other side by side that would normally be found arranged in a zonal fashion.
Then inside the resultant vortical formations of both water types, heavier water sinks
down at the boundary surface extending to more southern latitudes, while the lighter
water at the same time is lifted and extends further towards the poles. The sinking
process of the heavier waters apparently does not take place everywhere along the

The Tropospheric Circulation

657

658

The Tropospheric Circulation

extended more or less zonal boundary surface, but rather in form of individual mass
intrusions {quantum-like) at different places whereby as a consequence mixing is con-
siderably increased. The nature of the processes involved can be illustrated by putting
side by side successive stages of the oceanic state in a meridional section (Defant,
1941Z>), and one obtains thereby all the characteristics of the disturbances which occur.
The bottom topography in this part of the South Atlantic was earlier assumed (p. 435)
to be the cause of the wave-form current pattern appearing in the region of the sub-
tropical convergence (Fig. 187). It should be emphasized, on the other hand, however,
that the vortical disturbances originate on the shelf of the South American continent
between 45° and 35° S. far in the west, and from here extend as a continuous chain of
regular vortices throughout the entire area as far as the southern tip of Africa. This
source region or birth place, is the region where the denser water of the Falkland
Current meets the lighter water of the Brazil Current and where the tendency
for a vortex formation is extremely large. Here a strong solenoidal field is continuously
regenerated, which can be considered as the necessary condition out of which vortices
are formed and the disturbance field then stretches far out into the Atlantic.

A probable explanation of these wave-form disturbances can be derived by means of
the arguments put forward by Rossby and co-workers (1939) in a discussion of the
sinusoidal disturbances in zonal atmospheric air currents. In a wave-like disturbance,
which is superimposed on a pressure field that decreases to the south (Fig. 309,

P+2

W ^+

P+2

P + l

Fig. 309. Wave flow for a uniform towards south decreasing pressure field.

Southern Hemisphere) the water transport through the cross-sections A and C where
there is an anticyclonic curvature of the isobars will be greater because of the occurring
centrifugal force than that through section B where there is a cyclonic curvature. There
will therefore be a horizontal divergence and pressure fall between sections B and C
and a horizontal convergence and pressure rise between A and B. The wave disturbance
will thus move eastwards and since the centrifugal force is larger when the curvature is
greater the shorter waves will travel eastwards faster than the longer ones. In addition
to this effect, there will be a pure latitude effect which originates from the relation of
the geostrophic flow to the pressure gradient. Due to the Coriolis force the mass
transport across the section 5 in a lower latitude will be greater than that across sections
A and C in higher latitudes. This gives rise to convergence and pressure rise between
A and B. This latitude effect which is independent from the wavelength causes a west-
ward movement of the wave. Both effects are of the same order of magnitude and it is
easily understood that for a particular wavelength the wave disturbance will be

The Tropospheric Circulation 659

stationary. The mathematical basis extended by Haurwitz (1940) affords a relation
between the wavelength L, the latitudinal extent D of the stationary disturbance and
the velocity of the basic current, U, in the form

4772^ 1 +L^ID^'

whereby j8 = 8f/R8<f) = (2aj cos (f>)/R is the change of the Coriolis parameter / with
latitude and R the earth radius.

Analysis of wave disturbances in the South Atlantic convergence region gives an
average disturbance length at latitude circle 38° S. of 10-0° or 880 km. The latitudinal
extent averages 15° or 1650 km. With these values the velocity of the basic current U
is obtained as between 26 and 28 cm/sec. This means that the wave disturbance within
the zonal basic current (oceanic West Wind Drift) can be stationary only if such a
mean velocity towards East is present. Current charts show an average surface velocity
of 25-30 cm/sec. It is thus very probable that the stability of the stationary wave
system in the convergence region is due to an equilibrium state between the action of
the latitudinal dependence of the Coriolis force and the effect of the curvature of the
current trajectories on the horizontal mass transport. The strong solenoidal fields at
the boundary between the Brazil and the Falkland Currents may be responsible for
the formation of the eastward following series of vortical disturbances inside the general
oceanic West Wind Drift. If this is so then the topographical effect of the bottom
configuration will be only a supplementary effect which may intensify and probably
modify these disturbances.

Similar phenomena may also develop in the North Atlantic. In the oceanic strip of
the North Atlantic Current to the north of the subtropical convergence region there
are marked pulsations that also stand out clearly in the charts of the dynamic topo-
graphy of the individual isobaric surfaces and in that of the physical sea level. The
results of the International Gulf Stream Survey (1938) to the north of the Azores
enabled a study to be made of the oscillations in the current system in this particular
region. The oceanographic work of the "Armauer Hansen" in 1909, 1925 and 1935-6
in the Norwegian Sea off the coast of Norway (Helland-Hansen, 1934, 1939) showed
that vortices with vertical axes probably played an important role in the interior of
the Atlantic Current. They are also associated with considerable variations in mass
transport. It is rather obvious that such variations at fairly long intervals cause reactions
in the oceanic phenomena in the Arctic and take influence on climatic conditions in the
Scandinavian countries. At present, however, the investigation of these phenomena is
only at the very beginning and systematic and synoptic surveys are required in order
to obtain a deeper insight into the mechanisms involved. An unusual theory of the
variations of the surface circulation in the North Atlantic, especially of the current
branches off the coast of Europe, has been given by Le Danois (1934) in his Atlantic
Transgressions. He distinguished between three water types in the Atlantic: the
tropical, the polar and the continental. The latter has an extremely variable salinity
and remains at shallow depths in a relatively narrow band along the coasts. His
"transgressions" are periodic movements of variable amplitude carrying Atlantic
water of tropical origin, in temporary intrusions into water masses of polar and
especially continental origin. The water of the transgressive masses always has a

660 The Tropospheric Circulation

salinity greater than 35%o. From a large number of individual cases Le Danois has
attempted to derive definite rules according to which these trangressions move to the
north-east. These instrusions of Atlantic water into north-west European waters are
discernible only in their effects on the "continental" water masses over the shelf.
Here the warm transgressions at the surface over the continental plateau always are
preceded by highly saline transgressions in the deeper layers. The transgressions
appeared nearly always to follow the course of the valleys of the submarine relief.
The direction of spread is mainly to the north-north-east, so that the speed of this
spread of the intrusions is the less the more it deviates from this direction. By following
these phenomena in the sea off the coast of France for a large number of years Le
Danois has found certain periodicities in the occurrence of the transgressions, which
superimpose each other in the same manner as waves.

However, it appears difficult to follow the Le Danois theory of these transgressions,
since he uses several arguments quite contradictory to the established fundamentals
of dynamic oceanography (Schubert, 1935).

Chapter XX

The Stratospheric Circulation

1. Introduction

Beyond the polar convergence (oceanic polar front) towards the poles the oceanic
stratosphere reaches upward to the sea surface and is here subject to the full influences
of the atmosphere (radiation, evaporation, precipitation, freezing processes and
others). The water types continually formed by the climatic conditions here are
heavier, due to their low temperature and in spite of their low salinity, than the waters
of the adjacent convergence regions of the oceanic troposphere. Thus, in relation to
these latter water types they tend to sink, intruding beneath the oceanic troposphere,
until they reach a depth corresponding to their density. The sinking, strongly favoured
by the thermo-haline structure, reaches down to great depths. After sinking, the
almost horizontal spread of the water underneath of the troposphere causes a layered
leaf-like structure in the oceanic stratosphere. When this structure is sufficiently well
developed it is therefore possible to tell from it something about the path of spread
of the water masses and gain thereby an insight into the stratospheric circulation. This
is the method that has been used up to the present time in the study of the water
movements inside the stratosphere. In the absence of sufficient direct current measure-
ments, however, the results of such investigations were largely only of a qualitative
nature. Preparation of the observational data according to dynamic methods can
provide further insight into the nature of the stratospheric oceanic flow, but at the
present time only a few investigations of this type have been made. All these methods,
of course, give mean conditions only. Over large parts of the ocean, however, especially
for the deeper layers the basic prerequisite of stationary movements will be satisfied.
But aperiodic disturbances of shorter or longer duration and of greater intensity un-
doubtedly occur. By means of the observations available at present, and also due to
the manner in which they have been gained, it seems hardly possible to draw any
conclusions about the nature of these disturbances.

The surface layers of the oceanic stratosphere poleward (the polar fronts) are, of
course, subject to wind influence, so that also in the polar and subpolar seas wind-
driven ocean currents are generated. The complicated orographic configuration of the
continents in the Northern Hemisphere affects the nature of these surface currents
and exerts strong influence during their transformation into gradient currents. In
this way, piling up (Stau) phenomena play the principal role, and meridionally oriented
coasts in higher latitudes form excellent guiding channels for southward outbreaks
of the cold polar water masses. The zonal polar circulation obtains in that way
meridional components, so that on the eastern sides of polar land mass water flows
south, while on the western sides mainly water of subtropical origin flows north.

661

662 The Stratospheric Circulation

2. Polar Currents of the Northern Hemisphere

Phenomena similar to those found in the subtropical convergence region can be
expected also to occur at the polar convergences. These will be even more intensive
there, since a much sharper density difference exists between the adjacent water
masses. External factors will, at many places, cause the formation of vortices between
the warmer highly saline waters of subtropical origin and the cold weakly saline polar
waters. These travel along the boundary zone, continually forming and disappearing
and thus giving rise to a continuous mixing of the two water bodies. For these reasons,
in the Northern Hemisphere, the left-hand border of the polar currents is not sharply
developed and here polar waters and water masses of subtropical origin work into
each other. This is shown to be true for all currents, especially for the East Greenland
Current along its boundary region against the Irminger Current to the south of Iceland
and for the Labrador Current where it encounters the Gulf Stream.

Some insight into the processes involved in the vortex formation in the region of
interaction between two adjacent water masses, especially as found in this part of the
ocean, has been obtained from the almost synoptic surveys made by U.S. Coast Guard
vessels (see the bulletins of the U.S. Coast Guard, International Ice Patrol,
Washington).

The sea around Greenland (Greenland Sea, Labrador Sea, Davis Strait and Baffin

Bay) has been well surveyed oceanographically by numerous expeditions, and from

the entire data available it is possible to obtain an idea about extent and course of all

the currents. This is especially true of the East Greenland Current which can be follov/ed

along its entire course from the Denmark Strait to Cape Farewell and from thereon as

the West Greenland Current until it finally disappears (see Defant, 1936^ for

references). Little information is available on the East Greenland Current from its

origin near the Spitzbergen Rise to the Denmark Strait but there are appreciably more

data to the south of this strait. All cross-sections show a similar structure. The polar

water layer always has a cold core in which the temperature is almost at freezing point.

Figures 3 10 and 3 1 1 show two cross-sections through the East Greenland Current in the

Denmark Strait and off Cape Farewell, The analysis of 37 sections of this type through

the East and West Greenland Currents has enabled the course of the polar water

flowing around Greenland to be followed in detail. In Fig. 312 an attempt has been

made to show the boundary separating polar water from Atlantic water; in addition,

the average minimum temperature in the core layer of the polar water is indicated in

this figure which is usually at a depth of 80 m. The minimum temperature in the core

layer gradually rises from — 1-7°C in the Denmark Strait to about — 1-0°C at Cape

Farewell. Past the southern tip of Greenland, where the current turns sharply around,

the core layer rises towards the surface; its temperature increases rapidly and from

about 61° N. on is usually no longer negative. The East Greenland Current from the

Denmark Strait southwards where the width of it is more than two-thirds of the width of

the strait remains entirely over the shelf; where the shelf is broad the current is also

wide and where the shelf is narrow (for instance between 62° to 63° N.) its width is

very small and does not exceed 25 to 30 nautical miles. The lens of cold water forming

the current core at first extends well to the east, but becomes smaller towards south and

shrinks from the Denmark Strait to Cape Farewell under the impact of the warm

water of the Irminger Current. It is, however, still present and shows that the polar

The Stratospheric Circulation

663

Hdll2 Hdlll ^19 Hd63 0o4432 -^ZB Hd30 Hd3l Hd32 Hd33 Hd34Hd35

29 30 31 K 31 32 32 33 34 38

100

200

300

400

500

9519 Hd63 , pa4432 o528 Hd30 Hd3l Hd32 Hd33 Hd34Hd35

I 2 2 T 12 21 I 2 4 10

Fig. 310. Vertical cross-section through the East Greenland Current for the region of the
Denmark Strait at about 67" to 65" N.; below, temperature; above, salinity.

water is an uncustomary water type in the oceanic space under consideration and is
maintained only by continuous renewal. The intrustions of the Atlantic water occurs
in the form of vertical vortices which break through the polar front, broaden and
deepen and if the inflow weakens soon disappear. (Defant, 1930a; Bohnecke,
Hentschel and Wattenberg, 1930-32). From Cape Farewell the current bends
northwards still keeping also here over the shelf. At first the cold core layer is still
present but its temperature rises rapidly indicating stronger mixing with the Atlantic
Water penetrating northwards along the continental slope. From about 64° N. the
current weakens more and more and near the Davis Strait (about 66-5° N.) there are
only traces of the cold core layer found off the Greenland coast. In this region, in all

664

The Stratospheric Circulation

profiles, another core layer at about 80 m depth shows, which must clearly be fed from
the north-west by cold polar water that flows in through the Davis Strait with the
southward along Baffin Land directed current and finally joins the Labrador Current.

As yet no dynamic preparation has been made of all the available data for the East
Greenland Current. Topographies of the physical sea level and the isobaric surfaces
in this region are also contained in Fig. 271 (see also Fig. 200). The downward
slope of the isobaric surfaces from the Greenland coast towards the open sea is
quite large and shows clearly the entire system of the East and West Greenland
Currents. This current system can no longer be seen in the 800 decibar surface; the
stronger current intensity is thus confined to the top layers.

The main cause for the development for the East Greenland Current must lie in the
wind- and atmospheric-pressure conditions over the North Polar Basin. At all times
of the year due to wind and atmospheric pressure the water is driven eastwards and
water laden with pack-ice and drift-ice is carried towards the coast of north-east
Greenland. Here they find, supported by the wind turn towards south, a guiding
channel in the form of the Greenland coast. The pressure due to the piled up water
in combination with the action of the deflecting force of the earth's rotation produces
a southward gradient current. It could be expected that these cold weakly saline waters
on penetration into the warm but highly saline Atlantic water masses would soon be
dispersed by mixing. This is not the case and they still show, only slightly weakened,
as far as the southern tip of Greenland. They are maintained only by the continuous
advection of polar water from the north and by the climatic regime which maintains
the inland ice in Greenland. The polar climate generated by the inland ice, together

3 4 5 e

ico-

200

300-

400-

500

600

700-

800

200-

300

400-

500

600

700

800

Fig. 311. Vertical cross-section through the East Greenland Current somewhat north of
Cape Farvel (about 60' N.); left-hand side, temperature; right-hand side, salinity.

The Stratospheric Circulation

665

45° 40'

Fig. 312. Spreading of water masses of the East and West Greenland Currents derived from

35 oceanographic cross-sections. , limit between the east and west respectively

of Greenland Current and the Atlantic water. , limit of the core layer of the

Baffin water — 1-8'C: minimum temperature in the core layer of the polar water.

with the continous transport of cold inland air which spreads well out over the sea,
produces a belt around Greenland in which the temperature is lowered so much that
also there a polar climate prevails. Within this belt the East Greenland Current
maintains itself as a polar current as far south as 60° N.

An excellent monograph on the water masses of the oceanic region between Green-
land and North America with numerous temperature and salinity sections and velocity
profiles calculated on the basis of these sections is that by Smith, Soule and Mosby
(1937). For a general understanding of conditions here any of the cross-sections can
be selected from each current section since the main features are very similar in all of
them. Figure 313 shows these conditions in a cross-section through the Davis Strait.
The different water types moving through the strait are clearly shown, in particular
by the temperature distribution. Water with a temperature of less than — 1 °C keeps
well towards the Baffin Land side and forms the core of the Baffin Land Current;
its centre is found at about 100 m depth and here as shown by the velocity profile the
current direction points towards south. On the western side of the strait there is a
warm weakly saline top layer flowing northwards with a small velocity that represents
the last branching remnants of the West Greenland Current. There is a core of warm
and highly saline water at about 400 m depth; this is Atlantic water that moves
northwards within the lower layers of the West Greenland Current along the conti-
nental slope. Along the 600 nautical miles that this water travels in about 3 months
from Cape Farewell its temperature falls by 4°C and the salinity by 0-50°oo due to
mixing. The Labrador Current, after reinforcement by the inflow through Hudson
Strait, also keeps close to the continental coast and as in the case of the East Greenland

666

The Stratospheric Circulation

_ CM f^ ^3" in
1^ fv. r^ r^ ^-

30-91 30-85

Fig. 313. Cross-section of temperature (above) and of salinity (below) through the
Davis Strait ("Godthaab" stations 168-175, 17-19 September 1928).

32-10 32-34

34-42
32-53 34-41

0 40 80

I : 1 : !

miles

100

6-9

6-7
6-8

7-0

6-3 70
6-4 7-1

-r

■^\

ST^

O) "

200

-V.

a-y^

400

\ .3 44

3206 32-48 33-02

0 31-95 32-72 32-72 33-61

400

Fig. 314. Above :'cross-section Oi of temperature (to the left) and of salinity (to the right)
through the Labrador Current near to the Belle Isle Strait ("General Green" stations 1333-
1341, 7-8 August 1931). Below: Cross-section Pi of temperature (to the left) and salinity
(to the right) through the Labrador Current near White Bay ("General Green" stations

1229-1238, 6-7 July 1931).

The Stratospheric Circulation

667

Current the shelf forms the main path of the southward-flowing cold, weakly saline
water. Figure 314 shows this in a marked way and is characteristic of all the cross-
sections from Davis Strait to the Newfoundland banks.

A calculated dynamic topography of the sea surface relative to that of the 1500-
decibar surface based on a dynamic evaluation of all the observational data (1928-35)
is shown in Fig. 315. This gives some idea of the current conditions in the very upper
layers, since it should correspond rather well to the absolute topography. The trough-
like depression of the water level between Greenland and Labrador stands out parti-
cularly well in this figure, with an even narrower continuation reaching southward as
far as the southern end of the Newfoundland Banks. The strong concentration of the
dynamic isobaths and a high coastal water level off south-west Greenland indicates
the West Greenland Current and off the north-east coast of America the Labrador
Current, while the strong rise from the southern peak of the Newfoundland Banks
towards the north-east is due to the Gulf Stream. According to the topographies of the
600- and 1000-decibar surfaces, the strength of the currents decreases very rapidly
with depth. In the region of the Labrador Current there are differences in water level
of about 30 dyn. cm at the sea surface while over the same distance the difference in sea
level at 600 decibars is only 3 cm and at 1000 decibars is not more than 1 dyn cm.
These currents are thus typical density currents and are confined to the top layers.

Volume transports calculated from the velocity profiles for several different cross-
sections are given in Table 1 54 which also gives a rough budget for the water and heat
exchange amounts in the Labrador Sea. The pure gain in water is about 7-5 milhon
m^sec while the outflow along the Labrador Coast amounts to about 5-6 million m^sec.
Both values refer to a transport down to 1500 m depth. This gives a difference of 1-9
million m^/sec from which the authors assume that it is the water of the West Green-
land Current that sinks down to depths below 1 500 m and very probably flows out
of the Labrador Sea in the deepest layers. Figures for different seasons and for different
years vary considerably; for instance the transport of the Labrador Current was
1-31 milHon m^sec in 1930 and 7-60 in 1933. From this it must thus be concluded that

Table 154. Exchange of water and heat in the Labrador Sea
{after Smith, Soide and Mosby)

Exchange of

Water
X 10® m^ sec-^

Heat
X 10» kg cal

Inflow
West Greenland Current (average at Cape Farewell)
Baffin Land Current .....
Hudson Bay discharge .....

Total .

Outflow
West Greenland Current to Baffin Bay
Labrador Current (average South Wolf Island)

Total

50
20
0-5

7-5

4-6

5-6

17-5

-1-2

0-5

16-8

0-5
14-6

151

668

The Stratospheric Circulation

W 00 ^ 80 70 60 50 40 30 20 10

60 W

Fig. 315. Dynamic topography of the physical sea surface in the region of the Davis Strait

and the Labrador Sea, relative to that of the 1500-decibar surface (Mean for the period

1928-35, according to Smith, Soule and Mosby).

The Stratospheric Circulation 669

the out-flow from the Labrador Sea is subject to large variations dependent on a
number of diff"erent phenomena occurring at the sea surface of the polar regions (see
also KiiLERiCH, 1939).

Table 156 also contains a heat budget for the Labrador Sea. The heat gain amounts
to 1-7 X 10^ kg cal/sec. If the mean temperature of the waters which sink below
1500 m is taken as 3-2°C then the heat flux with the outflow mentioned above will be
about 6-1 X lO^kgcal. This then gives for the Labrador Sea a heat deficit of
4-4 X 10^ kg cal. It is not improbable that this heat deficit according to its magnitude
is totally compensated by the heat absorption of solar radiation in the water during the
summer. It can be calculated that of the total radiation from sun and atmosphere about
20 X 10^ kg cal reach the sea surface of the Labrador Sea. Of this then more than
40% (8 X 10^) is lost by reflection; the remaining 12 x 10^ kg cal goes to radiation,
evaporation and absorption. Since the radiation is probably not very eff"ective, about
two-thirds of this goes to evaporation and one third or about 4 X 10^ kg cal to
absorption. This quantity is of the same order of magnitude as the quantity given above,
but due to the uncertainty of the calculation this result should only be accepted with
reservations.

3. The Processes which occur at the Antarctic Convergence Zone

The causes for the formation of an Antarctic convergence within the broad oceanic
West Wind Drift of higher latitudes in the Southern Hemisphere were discussed on
p. 549. This discontinuity layer in the thermo-haline structure of the upper water
masses appears in the pressure field as a discontinuous step in the meridional slope of
the isobaric surfaces and the physical sea level (Fig. 253). This can also clearly be seen
in representations of the dynamic topography of the isobaric surfaces constructed by
Deacon (1937) according to the data obtained by the "Discovery" for the broad ring
of water surrounding the Antarctic continent. Figure 316 shows the dynamic topo-
graphy of the physical sea level (relative to that of the 3000-decibar level) for this
oceanic region. The downward slope of the pressure surfaces towards south at all meri-
dians is not uniform and a discontinuity extends all around the earth that makes the
meridional gradient much stronger in a belt coinciding with the Antarctic polar front.
This frontal zone is also shown to exist in the topographies of the isobaric surfaces
for larger depths; but corresponding to the much smaller gradient it is less strongly
developed in the deep sea.

From the analysis of a series of vertical sections between Antarctica and South
America (partly in the Drake Strait) as well as at 30° W. in the South Atlantic between
36° and 50° S. based on the observations of the "Discovery" Sverdrup (1933a) has
deduced the vertical circulation in the Antarctic Convergence Zone. Since conditions
around the Antarctic continent are very uniform, the results should be typical for the
whole of the circumpolar region. The essential details can be seen in the temperature,
salinity and oxygen sections at 30° W. shown in Fig. 317. According to all such meri-
dional sections and also according to those for the other oceans the water masses of
the upper layers south of the Antarctic Convergence sink down along the boundary
layer. In the salinity distribution this is clearly shown by a tongue of weakly saline
water. At the polar front at first the water sinks immediately down to 400 m and then
spreads almost horizontally to the latitude of the subtropical convergence region where

670

The Stratospheric Circulation

Fig. 316. Dynamic topography of the physical sea surface relative to that of the 3000-

decibar surface (according to Deacon). The figures are anomalies of the dynamic depths

referred to a homogeneous ocean of 0°C and a density a^ of 2800.

it sinks again rapidly to 800 m or more. The temperature sections show a tongue of
relatively warm water beneath the Antarctic water of the uppermost layers that forms
an intermediate layer between 400 and 800 m and must be interpreted as a returning
current flowing back towards south. Since this warm intermediate layer is found every-
where it seems to be a general phenomenon. Above it, in all sections (in summer), a
tongue with a lower temperature is found directed northwards at a depth of between
80 and 200 m. This stratification is no effect of a northward water transport but is
rather a remainder of the cooling which has been effective during the previous winter
(seePt. I, p. 137).

The Stratospheric Circulation

671

In the deep layers the salinity distribution indicates a deep flow from north to south
between 1800 and 3200 m and, beyond 40° S, gradually rising to 1000 m, while in the
far south just off the Antarctic continent the cold Antarctic water sinks to the bottom
layers of the ocean. The following sections of this chapter are devoted to these
processes.

Inside the water masses south of the convergence there is thus found in the upper
layers a vertical circulation that extends down to about 1000 m which occurs in an
anticlockwise sense when looking towards east. The uppermost layers are carried
northwards by the wind, sink down at the Antarctic convergence and form the main
constituent of the subantarctic intermediate current. Part of this water mass, however,
mixes with deep water and returns southwards in the Antarctic circumpolar ocean as
a warmer intermediate current. The top layers of sub-Antarctic water are rich in plant

675

1000

20CHD

3000

36° 38° 40° 42° 44° 46° 48° 50° 52° 54° 56°
675 673 671 668 666 663 661

1000

2000

3000

Fig. 317. Distribution of temperature (upper picture) and of salinity (lower picture) in a
vertical section at 30° W. from 34° S. to 58° S. in the South Atlantic Ocean (series measure-
ments of the "Discovery 11", end of April 1931, according to Sverdrup).

672

The Stratospheric Circulation

and animal organisms. Dead organisms sink downwards and decompose and therefore
the water of the returning intermediate current is also rich in phosphate. The lower
oxygen content in and just beneath the returning current indicates strong oxidation of
organic matter. Since only a part of the water transported in the uppermost layers
returns to the south there must be a compensating poleward component in the deep
layers in order to replace the cold polar waters sinking down in the very southern
latitudes along the Antarctic continental slope. Beneath the vertical circulation of the
upper layers there should therefore be a somewhat weaker one which rotates in a clock-
wise sense looking east. These vertical circulations are superimposed on a general
basic movement towards the east so that the resultant motion occurs in form of elon-
gated spirals. In these circulatory motions of the water the water properties are altered
in the upper layers by influences from the atmosphere above while in the lower layers
changes occur due to mixing. The water of the higher southern latitudes is thus made
up partly of water of the returning intermediate current and partly of deep water from
lower latitudes. The schematic block diagram presented by Sverdrup that is shown in
Fig. 318a shows the meridional components of motion in the Antarctic Circumpolar
Ocean.

A.C.

Fig. 31 8o. Schematic representation of the meridional circulation inside the Antarctic
Circumpolar Current (according to Sverdrup).

This concept of the circulation character occurring in these higher latitudes of the
Southern Hemisphere differs somewhat from the ideas expressed in elder investigations.
Merz and Wust (1928), for instance, interpreted the warm and highly saline water of
the intermediate layer of the higher southern latitudes only as the last traces of Atlantic
Deep Water reaching the sea surface in this region. According to Clowes (1933), this
water should be of Pacific origin and should reach the Atlantic only by way of the zonal
circulation. Both suppositions are only partly tenable. Sverdrup attempted to estimate
also the magnitude of the meridional velocity components, on the one hand, from the
shearing stresses of the wind leading to an estimate of the resultant total water trans-
port, and on the other hand, from the steady-state condition in the temperature field
of the returning intermediate current. For the upper layers a mean value was about

The Stratospheric Circulation 673

2-5 cm/sec. The time required to perform a single complete cycle in the upper vortex
with a horizontal axis in the area of Drake Strait thus amounts to at least a year when
the above mean velocity value is used. This transverse circulation is, however, undoubt-
edly stronger here than elsewhere in the Antarctic Circumpolar Ocean.

South of the oceanic West Wind Drift the physical sea level and the isobaric surfaces
rise again towards the Antarctic continent. An indication of this rise in the Atlantic
Ocean can be seen also in Fig. 316. Near the continent easterly winds prevail, and the
currents flow towards west. In this flow along the continent there will thus occur a
vertical circulation similar to that appearing in the oceanic West Wind Drift except
that it performs a clockwise rotation when looking east. There are indications of such a
circulation found in the observations of many Antarctic expeditions. In this connection,
Sverdrup also drew attention to the transport of lighter water by the wind towards
the Antarctic shelf where it is strongly cooled. The wind thus has a tendency to pile
up the lighter surface water against the shelf and produces stronger and stronger
solenoidal fields, which are of no consequence, however, since the water simultaneously
is cooled there. Both effects thus work in opposite sense and prevent the development
of strong solenoid fields and also of stronger currents which would otherwise be
formed solely by the action of the wind.

4. Dynamics of the Antarctic Circumpolar Current

It is of interest to investigate the extent, in a broad current which encircles the whole
earth, to which the wind stresses acting on the sea surface are balanced by frictional
stresses against the outer boundaries of the ocean basins. For most oceanic currents
the computed transports diff'er as was shown by Munk (1950), by a factor of not more
than 2 from the observed transports. Munk and Palmen (1951) have made a similar
calculation for the Antarctic Circumpolar Current. They considered the Antarctic
Circumpolar Current as an eastward flow on a plane tangential to the earth at the
south pole. The flow is induced by the constant eastward winds and depends only on
the distance r of this plane from the pole. The balance between the wind stress T and
the lateral friction is expressed by the relation

where A^ is the lateral kinematic viscosity and M is the eastward mass transport across
a normal vertical plane of unit width extending from the sea surface to the sea bottom.
For a solution in which M vanishes at the Antarctic continent {r = r^), and at some
other latitude (r = r^) the total mass transport of the flow will be

18AV^ '« r, + ro'"/-oy
Putting r = 2 dyn cm-^, A^ = 10^ cm^ sec-^ and /"o = 70° S., r^ = 45° S. one obtains
M = 5 X 10^*^ g sec"^ whilj the observed transport is at least 1-5 x 10^* g sec"^.
This discrepancy is not materially altered on taking spherical co-ordinates or allowing
for the variation of the wind with latitude. The transport M is inversely proportional
to Ah and only values of 10^°t or more can give an agreemenwith the observed facts.
Values of Ah as large as this are. however, improbable. Munk and Palmen attempted
to reconcile the two values by taking into account the friction at the bottom especially

M - I Mdr= ,^^ I ri^ - r^^ - :^^ In -| . (XIX.2)

674 The Stratospheric Circulation

there where the major submarine ridges lie as transverse obstacles in the path of the
current. If the wind stress should be completely balanced by the frictional stresses
along the sea bottom, then the Antarctic Circumpolar Current must extend deep
enough to reach the sea bottom. It is certain from the vertical oceanic stratification in
these latitudes that the current reaches down to very large depths; this is clearly
indicated by the dynamic topographies of the individual isobaric surfaces. However,
the velocities decrease very rapidly with depth and at depths of more than 4000 m
the flow intensity of the Antarctic Circumpolar Current is extremely small. Corres-
pondingly, the frictional stresses at the sea bottom will also remain rather small. By
making the most favourable assumptions Munk and Palmen showed that the retarding
pressure of the submarine ridges against the deep current might still be able to balance
the wind stress on the surface.

HiDAKA and Tsuchiya (1953) have recently taken up the problem again and
attempted to find a hydrodynamic solution. From the equations of motion and the
continuity equation with the corresponding boundary conditions they derived for
planar co-ordinates, a complete solution in the form of infinite series giving the total
mass transport, the surface slope and the vertical velocity distribution. Their calcula-
tions using some arbitrary numerical values of the lateral and vertical eddy viscosity
{Ah and y4„) give the same results as those of Munk and Palmen. For A„ = 2 x 10^
and Ah = 10^" cm-^ g sec"^ they found a total mass transport of 9-3 x 10^^ g sec~\
a surface slope of 3 m per 25° lat. and directions and strength of the currents in good
agreement with those observed. But also in this case choosing values of ^4^ less than
10^^ would give impossible conditions. In a more recent treatment of this problem
Takano (1955) introduces a special vertical and meridional density distribution corres-
ponding approximately to the observed ones. The rather complicated mathematical
solution led to the following conclusions: if the Ekman frictional layer is disregarded
then the geostrophic approximation can be safely applied for the small velocities near
the sea bottom. However, in order to obtain agreement with the observed values of
the surface velocity, of the surface slope, of the density diff'erences at the sea surface
and of the mass transport, it is necessary to take ^4^ = M x 10^". This is again the
same large value that was found to be a necessity in the investigations mentioned
before.

There must thus be yet another source of energy dissipation in order to have a
complete balance in the sense put forward by Munk and Palmen between wind stress
and frictional stress. This can probably be obtained by taking into account the effect
of the boundary friction, not only at the sea bottom but rather along the extended
continental slope of the Antarctic continent which was previously neglected. An essen-
tially different explanation of the dynamics of the Antarctic Circumpolar Current has
been given recently by Stommel (1957). While Munk and Palmen and all others who
have treated the problem regarded the Antarctic Ocean as an example of an ocean
without meridional barriers for which a Sverdrup type solution could not be con-
structed, Stommel believed that while the circumpolar ocean was indeed a continuous
ring of water around the earth, it was so strongly narrowed at Drake's Passage between
Grahamland and the southern tip of South America that a pure zonal flow could
hardly develop in this section. On this basis the Antarctic Circumpolar Current is
amenable to treatment by the Sverdrup theory and is essentially frictionless except

The Stratospheric Circulation 675

in a short section after its passage through Drake's Passage. The entire energy dissipa-
tion and all the other disturbances occur at this point; in all the other sections of the
current course it is a simple frictionless geostrophic current.

Stommel developed a simple model (Fig. 318Z7,(af) consisting ofa homogeneous ocean
of uniform depth surrounding a schematic Antarctic continent and only at one place
(indicated by the heavy radial line) a barrier closes Drake's Passage completely. The
zonal wind system assumed is also shown in Fig. 3186,(<^) with trade winds from the
equator to 30° S., westerlies from 30° S. to a little over 60° S. and further south a nar-
row zone of easterlies. The Ekman drift current transport is northwards in the
westerlies and southwards in the easterlies. Therefore a divergence zone exists between
about 55° and 50° S. and a convergence zone further north. Since there is a complete
barrier it is not difficult to maintain a wind-driven circulation. The meridional
components of this circulation are indicated by arrows in Fig. 3186,(fl) and the entire
circulation is shown in Fig. 318Z),(^)- At the western coast of the ocean (the eastern side
of the barrier) an intense western boundary current will be set up and this simple cir-
culation will be characterized by two immense gyres around the earth parallel to the
latitude circles. Stommel calculated that the transport in the southern gyre would be
somewhat more than 100 x 10^ m^ sec"^, and somewhat less in the northern gyre.

In fact, however, the northern gyre is broken up by the African and by the Australian-
New Zealand land mass. If now the barrier between South America and the Antarctic is
broken in the manner indicated in Fig. 3 1 86, (c) then the transport lines will run through
this opening and a circulation to the east will develop at the southern rim of the
barrier. The flow through the passage still remains unexplained but without doubt
models can be devised in order to describe it. Stommel's explanation of the dynamics
of the Circumpolar Current is quite different from the previous explanations. He also
attempted to make this explanation more plausible by embedding this current system
under consideration into the system of the sub-Antarctic-Antarctic circulation.

5. The Sub-Antarctic Intermediate Current

The most important facts concerning the spread of the subpolar Antarctic inter-
mediate water as far as they can be deduced from the distribution of the oceano-
graphic factors have been described already in Pt. I, p. 173. This water type forms the
uppermost part of the oceanic stratosphere. The sinking at the polar convergence is
shown by all meridional salinity sections (see Pt. I; Fig. 62 for the Atlantic, p. 147,
Fig. 75 for the Indian Ocean and Fig. 76 for the Pacific, p. 172). The fact that this
process at the Antarctic convergence (see p. 669) occurs with about the same intensity
all round the earth indicates that at all places the sinking and the subsequent spread
of this water type are caused by the same factors.

In the Northern Hemisphere the morphological configuration of the continents
interferes with the formation of an Arctic intermediate current and traces of it are
found only along the western side of the Atlantic. The weakly saline intermediate
current in the Atlantic is consequently not symmetrical about the equator and we
may only speak of a sub-Antarctic intermediate current here. In the Pacific the
northern current branch is almost as strong as the southern one and therefore in
the region of the thermal equator (6° to 8° N.) very similar water types come in contact
with each other. In the Atlantic the Antarctic branch is, however, so strongly

676

The Stratospheric Circulation

developed that it extends past the equator and can be traced almost as far as 20°
N. It is noteworthy that the thickness of the intermediate water at first is about 1000 m
and later on diminishes in wedge-form, and that it is found across the entire width
of all cross-sections through the ocean (see Pt. I, Fig. 77 p. 174).

A detailed analysis of the sub-Antarctic intermediate current in the Atlantic —
which is the only ocean for which this is possible at the present time — using the core
layer method and the [r5']-relationship has been given by WiJST (19366). By a deter-
mination of the percentage with which the original water type can be found south of
the Polar Front at each place in the entire space, and how much of it has been lost due

Fig. 3186. {a) The schematic southern ocean. Antarctica is the full black circle. The meri-
dional barrier projecting out from Antarctica is represented by the full heavy black line.
The schematic wind system (purely zonal) is depicted by the heavy arrows on the lower
left. Latitudes of Ekman convergence and sinking at the surface are indicated by 0,
latitudes of divergence and upwelling are indicated by ®. The direction of the required
meridional geostrophic flow is indicated by thin radial arrows.

(Jb) Transport lines of the solution for the model depicted in Fig. 3186,(«) The western
boundary currents are to be interpreted schematically.

(c) Hypothetical form of the solution, that results from rupturing the American-Antarctic
barrier in such a way as to permit water to flow throughout, to obstruct all latitude
circles (according to Stommel 1957).

77?^ Stratospheric Circulation

677

90' 80° 70° 60° 50°

20° 10° 0" 10° C0° 3C

W 120' 110° iOO° 90° 80° 70° 60° 50° 40° 30° 20° 10° 0° 10° 20° 30° 40° 50° 60° E

Fig. 319. Absolute topography of the 800-decibar surface (smoothed representation).

(Dynamic isobaths north of the subtropical convergence region drawn from 1 to 1 dyn cm,

otherwise from 5 to 5 dyn cm.)

678 The Stratospheric Circulation

to mixing, one obtains a rather good insight into the mixing process going on in the
total space of spreading (with reference to these conditions see Pt. I, p. 212 and
following pages and particularly the Figs. 100-102). In general, the diagrams indicate
a uniform spread towards the north taking place over the whole cross-section almost
immediately after the sinking at the polar convergence, but further north there is a
preference for the western half of the ocean which must be due to the effect of the
Coriolis force. Here close to the South American continent the spread possesses
current character. The entire width of the layer across the total ocean gets its supply,
then from the western side by lateral turbulence and by occasional occurring large
intrusions but the salinity distribution shows only the final stage after lateral mixing
has been effective and does not give information about the nature and way with which
the lateral mixing process operates.

Since a current is formed on the western side along the South American continent
these processes can be regarded as a case of free turbulence (Defant, 1936c) and the
ratio [exchange: velocity] can be determined along the entire spread of this water
type. This then gives some idea about the current character of the spread of the
Antarctic intermediate water. In order to find the pressure forces that give rise to this
water transport it is necessary to determine the dynamic topographies of the isobaric
surfaces at these depths. The absolute topography of the 800-decibar surface which
corresponds north of 40° S. closest to the core layer of the sub-Antarctic intermediate
water is shown in Fig. 319 for the region of 20° N. South of 40° S. the zonal course of
the dynamic isobaths indicate the downward extension of the large Antarctic Circum-
polar Current flowing eastward; but at this depth the meridional pressure gradient is
only half of that observed at the sea surface. Also, the broad high-pressure ridge in the
subtropical convergence region is present only with a somewhat diminished intensity
and in the convergence regions still vortical disturbances appear extending down to
these depths.

North of the high pressure ridge the isobaths run also from east-north-east to west-
north-west, but beyond 25° W. they turn towards the north and finally run along the
South American continent as far as Cape San Roque. The pressure gradient here is
thus directed towards the east but this gradient does not extend very far out from the
coast; the broad area from about 20° S. to 20° N. as far as the African coast shows
almost no gradient. Already downward from 500 m the water movements in this large
region must be extremely weak and there is no indication whatsoever of a circulation.
The water displacement corresponding to the absolute topography (see Fig. 320) on
the northern side of the subtropical disturbance zone in the Southern Hemisphere is
directed first to the west-north-west and then to the north-west and finally extends
in a narrow band along the South American coast as far as the West Indies and con-
tinues into the Gulf Stream region. The velocities everywhere remain small, between
6 and 12 cm/sec in the core layer, falling rapidly to weak intensities towards the eastern
edge.

An analysis of the salinity distribution in the Intermediate Current gives values for
the ratio [exchange :velocity] of 0-8 to 2-3 at the upper and lower edges, respectively.
This leads to exchange coefficients of about (5-10 g cm"^sec"i) which is in good
agreement with the order of magnitude found by other methods at such
depths.

The Stratospheric Circulation

679

680 The Stratospheric Circulation

6. The Polar Bottom Current

The second water type originating at the sea surface of the Antarctic ocean is the
Antarctic Bottom Water. It is formed all along the Antarctic continental shelf and
especially in the area of the Weddel Sea which is the place of formation for this coldest
and thus heaviest water type; it sinks along the continental slope down to the greatest
depths and extends northward following the bottom topography of the ocean as an
Antarctic Bottom Current. As it spreads it is subject to continuous mixing with the
water masses above. Its spread is hindered by transverse ridges which the current
must pass and limits are set to spread by meridionally oriented rises; the deep passages
through these zonally and meridionally oriented ridges thus form important guiding
channels for the bottom currents. The extension of antarctic bottom water in the
individual oceans as deduced from the thermo-haline structure has been described in
detail during the discussion of the temperature distribution in the bottom layers of the
ocean, so that the reader is only referred to this here (see Pt. I, p. 149). The spread of
the bottom water is shown in Plate 4 by lines of equal potential temperature and from
these the course of the bottom currents can be readily followed.

The generation of bottom water in the Antarctic is so enormous that the same
process in the Arctic is by comparison quite insignificant. In the Atlantic one can
hardly speak of any proper Arctic bottom current, since the high upward extending
ridges between North America, Greenland, Spitzbergen and further to the south
between Greenland, Iceland, the Faeroes and Scotland almost completely block the
outflow of bottom water from the Arctic Basin. Bottom water with a characteristic
potential temperature of between —0-2° and — 1-5°C passes over the above mentioned
submarine rises into the open ocean in only very small amounts.

Recent investigations of the flow near the bottom across the Iceland-Faeroes Ridge
have been made by Dietrich (1956, 1957). All the five cross-sections over these rises
have shown that the warm North Atlantic Water and the cold sub-Arctic water
are in contact over the ridge forming a narrow frontal zone. The heavy sub-Arctic
water lying underneath the lighter north-east Atlantic water always covers a large part
of the summit plateau of the Iceland-Faeroes Ridge, and sinks down immediately on
its western side because of its higher density keeping thereby close to the slope. In
spite of mixing with warmer water of smaller density its density remains still higher
than that of the surroundings, and consequently it sinks to form the bottom water in
the north-east Atlantic at depths below 3000 m. The velocity of this downward
directed bottom current on the western side of this ridge can be determined using a
formula given by Defant (1955) and results to about 35 cm sec"^ For a thickness of
the sinking water of 50 m and with a total width of the passage of 150 nautical miles
the water transport will amount to about 50 x 10^ m^ sec"^ Like a waterfall these
waters flow out in individual bursts and may be observed at any time of the year at
the Iceland-Faeroes Ridge. Oceanographically they have a greater importance than
the sinking movements caused by winter cooling over the shelf of the Bay of Biscay
and elsewhere along the continental shelf and slope which can occasionally be observed
(see Cooper and Vaux, 1949 and Cooper, 1952).

The main mass of North Atlantic Bottom Water thus originates outside the Arctic.
WiJST (1943) termed this water type with a potential temperature of between 1° and
2°C as the sub-Arctic bottom water and the current fed by it the ^'sub-Arctic

The Stratospheric Circulation

681

bottom current.'''' This sub-arctic bottom water comes mainly from two source
regions:

(1) from the north-western Labrador Basin where the colder bottom water with a
temperature of less than 1-2°C is formed (Wattenberg, 1938; Smith, Soule
and MosBY, 1937) and

(2) from a region of formation extending all along the 3000 m depth of the south-
east Greenland continental slope into the inner angle of a bay; this source was
already referred to by Nansen (1912).

From these two main centres the sub-Arctic bottom water spreads out towards more
southern regions. Influenced by the bottom topography, this spread, however, keeps
close to the western side along the foot of the continental shelf off the Labrador coast
as far as 50° N. A Labrador submarine rise here prevents its further southward spread.
The second centre of formation in the Irminger Sea is obviously less productive; since
already in about 55° N. this water type has mixed with warmer waters and has lost its
characteristic cold temperature.

A small Arctic bottom current also occurs in the Pacific; cold bottom water in
moderate amounts penetrates over the boundary rises of the Okhotsk Sea into the
open ocean. However, this is likewise only of sub-Arctic origin and its productiveness
remains small.

The ratio [exchange: velocity] can also be derived from the analysis of meridionally
oriented temperature and salinity sections (see Pt. I, p. 153) and stream lines of the
water transport can be constructed in order to obtain a representation of the current
course in its core (Fig. 321). The stream lines follow closely the bottom topography.
Over the crests of the ridges values of the above ratio lie between 2 and 3, in the depres-
sion between 5 and 6. For the same values of exchange the current intensity shows a
proportion of about 2-5:1. Wattenberg (1935) by keeping track of chemical processes
at the sea bottom and in the layers just above it found an exchange of about
4 cm~^ g sec"^. With this value, the velocity of the bottom current on the western side

Brasilian Basin

9000 10000

Guyana-Basin

14 000 15000 16000

3000-

D 5000

40» S 3

Fig. 321. Stream lines and values of the ratio between exchange and velocity in the core of

the Antarctic Bottom Current in the Atlantic Ocean (evaluated from the temperature and

salinity distribution in a longitudinal section of the Western Atlantic Trough).

682

The Stratospheric Circulation

of the Atlantic should be of the order of 0-5-2 cm/sec. In order to flow through the
distance from 50" S. to the equator, Antarctic waters would thus require about
10-30 years and would have lost 40% of its characteristic water properties on reaching
the equator. Variations in these properties occurring at a certain moment in the area
of formation of the water types could only be noticed in the bottom layers at the
equator after appreciably long time and with a considerably diminished intensity.

WiJST (1957) has recently made a dynamic investigation of the "Meteor" profiles
and has thereby extended the determination of the absolute topography of the physical
sea level and the isobaric surfaces made by Defant to the layers between 2500 m and
the deep-sea bottom. He based his computations on the topography given by Defant
for the dynamic reference surface (p. 496) and continued the calculations from this
surface to the sea bottom. These topographies were used to determine the velocity
components at right angles to the profiles. Figure 322 shows the resultant chart of the

xS-^S

i-tW.

.♦ v^-y

T ^;^m^u.-. hi::- />:V"7'

\ *s\ ■'■■

Northward current component
^I^Southword current corr.ponent

■A Axis of Antarctic bottom current
— Core of Antorctic bottom water

75° 60° 45° 30° 15° 0° 6°

Fig. 322. Current distribution in the Antarctic Bottom Water of the Atlantic Deep Sea (in a

depth of more than 3500 m) computed from the mass distribution taking as a basis the

reference level of Defant (according to WiJST, 1957).

The Stratospheric Circulation

683

bottom currents. The Antarctic bottom current in the Southern Hemisphere shows
measurable velocities (>3 cm sec~^) only close to the western side of the West Trough,
that is, at the foot of the continental slope and about 1000 m above the level of the
proper deep-sea bottom. With few exceptions only very weak velocity components
were found in the east. These results derived from dynamical computations agree well
with the above described ones. With these new velocity values the water masses of the
bottom would need about 5-5 years in order to travel from the southern rim of the
Argentine Basin (48° S.) to the northern rim of the Brazilian Basin (5° S.).

7. The Deep Currents in the Middle Part of the Oceanic Stratosphere of Individual
Oceans

In a fully symmetrical ocean there would be in each hemisphere a subpolar inter-
mediate current in the uppermost part of the oceanic stratosphere and a polar bottom
current in the lowermost part of it. These water transports directed towards the
equator for reasons of a compensation require an additional poleward water trans-
port in the middle part of the stratosphere. These compensation movements are
called the "deep currents" of the oceans.

In this way the scheme of the meridional components of the stratospheric circulation
(Fig. 323) thus consists of two closed circulations in each hemisphere; one circulation
in the upper part of the oceanic stratosphere containing the intermediate current and
the upper half of the deep current and moving in a clockwise sense when looking east
in the Northern Hemisphere, and a second circulation in the lower part of the oceanic
stratosphere that includes the polar bottom current and the lower half of the deep
current and moves in an anticlockwise sense. It should be borne in mind in looking at
Fig. 323 that only the meridional flow components of the two circulations are shown
which are always weaker than the zonal ones.

The rather varying character of the polar components in the actual oceans gives

1000
2000
3000
4000
5000
fionn

PC E C P

V^V^-:- S- -^ -^ - -\C - ^ ^ - ^ -'A^'u

i i ' 1-- -J -1 ; -t - 1 ,..,(, 1 ! 1 1

60°

40°

E0°

0°

20°

40°

60°

Fig. 323. Schematic representation of the meridional components of the oceanic circulation

in a symmetrical ocean. -« < , circulation of the troposphere; •^— , subpolar intermediate

currents; < , polar bottom currents of the stratosphere; < , mean deep currents of

the stratosphere; , limit between oceanic tropo- and stratosphere; P, polar front

(polar convergence); C, subtropical convergence; E, equatorial counter current.

684 The Stratospheric Circulation

rise, of course, to large differences in the development of the deep currents. The closest
approach to the ideal case pictured in Fig. 323 is found in the Pacific. The meridionally
oriented sections, although they are based on insufficient data and often do not reach
right to the bottom show the approximately symmetrical arrangement of the
subpolar intermediate currents about the equator. Warm water sinks in the convergence
region of both these currents as it is required in Fig. 323 ; such downwards motions are
indeed indicated by a downward bulging of the isothermal layers in the meridional
temperature sections. At greater depths the deep layers of the Pacific are almost uniform
and there is no special differentiation to indicate any particular motion (Wust, 1929,
\9Z0b). This is supported also by the absence of any temperature inversions which are
very characteristic of the Atlantic and the Indian Ocean.

The marked asymmetry of the polar components in the Atlantic Ocean due to the
almost complete absence of the Arctic current branches gives rise to a strong develop-
ment of the southward directed North Atlantic Deep Current. This provides the only
compensation here for the Antarctic water carried north by the intermediate and
bottom current. Disregarding at the moment the water layers from about 1000 to
1500 m between 50° N and 20° N. (particularly on the eastern side) the oceanic spaces
underneath are filled with relatively salinity- and oxygen-rich waters. The structure of
these waters indicate by their vertical structure a sub-Arctic origin. Its principal
characteristic is the oxygen content of the core layer and the distribution of this shows
clearly its origin from the area east and south-east of Greenland and from the boundary
zone between the East Greenland Current and the Irminger Current south-west of
Iceland as well as from regions in the north of the Labrador Sea. These are the same
regions that form the source of the sub-Arctic bottom water (p. 680). WiJST termed
this sub-Arctic bottom water as the "Lower North Atlantic Deep Waters" as opposed
to the "Middle Deep Water" occurring above. In these regions mentioned above the
almost homogeneous structure of the sea during autumn and winter allows the surface
waters to sink to great depths forming there the source of the more or less horizontal
southward water transport between 1 500 and 2500 m depth.

In the "Meteor" cruise made in late winter 1935 the kind of conditions were found
along a profile south of Greenland which are required to allow the autumn and winter
convection to proceed to great depths. The oxygen distribution along this profile
(Fig. 324) clearly shov/s this downwards tendency of the surface layers (Wattenberg,
1938). Below 1000 m the source for the middle North Atlantic Deep Water is formed
here. When this water moves further to the south the transport obviously keeps
closely to the western side of the ocean due to the influence of the Coriolis force, but
even after crossing the equator it still prefers the western side and the effect of the
Middle Atlantic Ridge is clearly noticeable. The upper layers of this southward water
movement show the effect of mixing with Mediterranean water (see later) since the
[TS]-relationship for the core layer at 35° N. shows a definite reversal point (see Fig.
325); apart from this the curve as far as 50° S. is almost a straight line and indicates
gradual mixing with the water types above and below. Beyond 30° S. these waters
enter the deep-reaching circumpolar flow of the very southern latitudes and under the
influence of this are deflected to the east. The pressure conditions in the core layer of
the North Atlantic Deep Water are best indicated by the topography of the 2000-
decibar surface. Figure 326 shows immediately that the main course of the isobaths is in

The Stratospheric Circulation

685

Fig. 324. Distribution of oxygen (in percentage) in a section from the southern tip of
Greenland to the Great Banks of Newfoundland (according to Wattenberg).

Fig. 325. Standard curve of the [r5]-relation in the core-layer of the mean North Atlantic

Deep Water.

its main features in agreement with the spread of this water type deduced from the
thermo-hahne structure.

The source of this water transport is in the north-west of the Atlantic and from here
it flows southwards principally in three branches (see Fig. 327). The western branch
keeps close to the North American continental slope, passes through the North Ameri-
can Basin and enters into the Southern Hemisphere to the east of the Antilles. The
middle branch follows the eastern slope of the middle Atlantic Ridge as far as 5° N.

686

The Stratospheric Circulation

120 » W 00°

40" E 60°

Fig. 326. Absolute topography of the 2000-decibar surface in a somewhat smoothed
representation (dynamic isobaths are drawn from 2 to 2 dyn cm).

The Stratosphere Circulation

687

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688 The Stratospheric Circulation

and then breaks up into vortices. The third much weaker branch meanders along the
East Atlantic Trough past Madeira to the Canaries and the Cap Verde Islands and a
side branch of it seems to enter the Guinea Bight. The course of the first two branches
under the influence of the Coriolis force used apparently the bottom morphology as
guiding limits for their spread. The westernmost and most important branch keeps also
in the Southern Hemisphere at first close to the continental slope until about 25° S.
and then bends towards east-south-east and fills from here as a broad water transport
the total oceanic space between 25° S. and 40° S. Finally, it passes south of Africa into
the Indian Ocean. The velocities in the Northern Hemisphere branches of the current
are seldom more than 2 cm/sec. Where the current concentrates along the South
American continental slope it reaches about 3-4 cm/ sec until 15° S. and at Cap San
Roque it reaches maximum speeds of 8-12 cm/sec before falling off to 0-5-1 -5 cm/sec
further south.

The spread of middle North Atlantic Deep Water as deduced from the oxygen
content of its core layer is shown in the left-hand chart of Fig. 327 ; the arrows in this
figure indicate the principal branches of spread determined from the dynamic topo-
graphy of the pressure surfaces. The agreement between the results of the two methods
is remarkable. Sverdrup (1930) has given a diagram showing the deep currents in the
southern part of the South Atlantic based on the "Carnegie" observations that fits
well in the topography of the 2000-decibar surface.

WiJST (1957) has calculated the corresponding velocities at right angles to the
"Meteor profiles" for the current course of the Atlantic Deep Water in the area
between 10° N. and 30° S. The distribution of these velocity components is presented
in Fig. 328 and shows obviously good agreement with the distribution in the right-
hand diagram of Fig. 326. In the core the velocity (reduced to the "true" direction)
is now 9-2 cm sec~^ with individual values varying between 2-1 and 17-4cmsec"^.
It should especially be noticed that also here the flow is concentrated towards the west
just off" the American continent while the eastern parts of the oceans are completely
inactive.

In the Indian Ocean a deep current stands out between 2000 and 3000 m marked
by a highly saline deep layer and a pronounced temperature inversion. Its strong
development is due primarily to the large density differences between the equatorial
and polar water masses which are continuously renewed by the supply of salt from the
Red Sea and the Persian Gulf (Pt. I, pp. 183 and 529). A deeply penetrating detailed
analysis of some oceanographic series observations in the Indian and Pacific Oceans
has been made by Helge Thomsen (1933, 1935). From the [rS'J-diagrams it appears
rather doubtful whether there is actually a deep current in the Indian Ocean between
2000 and 3000 m similar to that in the Atlantic. On the other hand, the Intermediate
Current and the Bottom Current are well developed as well as the effects from the
Red Sea are easily followed far to the south.

8. A Survey of the Water Transports in the Individual Layers of the Atlantic Ocean

The total amounts of the water transport in meridional direction in the South
Atlantic total space (between 5°S. and 35° S.) which Wiist has derived from mean
velocity values calculated from the individual profiles of the "Meteor" expedition are
of great interest. The most important results arc summarized in Table 157. The figures

The Stratospheric Circulation

689

Fig. 328. Current distribution in the lower Atlantic Deep Current (3000 m depth) com-
puted from the mass distribution taking as a basis the reference level of Defant (according

to Wust, 1957).

given in this Table show considerable scattering due to random errors and inaccuracies
in the basic data. Nevertheless, they give a rather good idea of the budget of the water
transports in the South Atlantic space which is valuable in many respects. The final
budget of the meridional transports (current amounts) is practically perfect with a
discrepancy of only O-I million m^ sec~^ A complete balance between northward and
southward transports in each of the two troughs cannot be expected. In the Western
Trough the North Atlantic Deep Current with 27-5 million m^' sec^ towards the
south is the main circulation component ; this is very largely confined to the narrow
strip along the South American coast. The transport in the uppermost part and with
the Bottom Current together is only 9-0 million m^ sec^ In the Eastern Trough the
transport towards the north in the bottom and deep currents is exceedingly small.

690

The Stratospheric Circulation

There is no current here which can be continuously followed through carrying water
in large quantities to the south. Probably only very weak spreading and mixing
processes operate here in variable direction. The deep sea circulation of the Western
Trough is thus dominant and sets the basic pattern for the whole of the South Atlantic
oceanic space.

Table 155. Mean values of the meridional water transport in the total space of the
South Atlantic Ocean {between 5° S. and 35° S.) given in units 10^ m^ sec ^

Current constituents

Water transports throughout
entire width of the ocean

Through the
Western
Trough
towards 1

Through the
Eastern
Trough
towards

Towards the Towards the
north south

north

south

north

south

Sea surface current "1
Deeper currents >
Intermediate current J

Deep current
Bottom current

22-7
3 0

25-6

70
20

27-5

15-8

} 4-8

—

9. The Effects of the Subtropical Adjacent Seas on the Deep Sea Circulation.

Analysis of series measurements in mid-latitudes of the eastern North Atlantic
led already at an early stage to the recognition of a warm highly saline water type with
little oxygen content, the principal characteristics of which point towards the Straits of
Gibraltar which can therefore be considered as effects on the waters of the Atlantic,
of the water flowing out of the European Mediterranean. The significance of "Mediter-
ranean water" in the Atlantic deep-sea circulation was first pointed out by Jacobsen
(1929).

A detailed investigation and review of the phenomenon was then given by WiJST
(1936) in the "Meteor" Report; he termed this water type "upper North Atlantic
Deep Water". It is characterized by its high salinity which is in sharp contrast to the
Antarctic intermediate water above it. Off Spain the core layer can be found at about
1000-1250 m and lowers down towards the equator reaching a depth of 2000 m
between 10° S. and 20° S. From the salinity distribution in the core layer it is immediately
obvious (see Fig. 329) that the spread takes its origin from the waters off Spain, and
that it obtains its high salinity content of 36-4%o or more by way of the Mediterranean
water flowing out through the Straits of Gibraltar in the lower layers (p. 529 et seq.,
see also, pt. I, p. 182). This water sinks to about 1000 m where it finds a corresponding
density and then spreads out in a fan-like fashion under the action of turbulence and
Coriolis force. Figure 330 impressively shows the great distances to which still an effect
of the Mediterranean Water can be traced. It extends northwards past 50° N. and it
reaches particularly pronounced directly across the entire Atlantic as far as the Ameri-
can coast. Towards the south the last traces can be followed even to the higher
latitudes of the Southern Hemisphere. The percentage of Mediterranean water present
at any point can be determined from a standard curve for the [rS'J-relationship in the

The Stratospheric Circulation

691

iOO" W 80

Fig 329. Spreading and depth of the upper North Atlantic Deep Water (Mediterranean
Water) The thin dashed Unes indicate the depth of the core layer in metres (according to

Wiist).

692

The Stratospheric Circulation

core layer of this upper North Atlantic Deep Water (Fig. 330). In the western part of
the North Atlantic there is still a content of 25-30%, at the equator 20-18% in the
South Atlantic the Mediterranean content gradually falls to below 2%. The form of
the [r^l-relationship which is nearly a straight line indicates that the changes in the
core layer are due essentially to a simple mixing process.

The great effect of the water flowing out from the Straits of Gibraltar on the compo-
sition of the water masses in the Atlantic is at first sight astonishing. A rough calcula-
tion shows, however, that it is of the right order. According to Schott (1939), about

366

Fig. 330. Standard curve of the [TiS] -relationship in the core layer of the upper North
Atlantic Deep Water (Mediterranean Water).

52-000 km^ of water a year flows out from the Mediterranean into the Atlantic. For
a mean velocity of spread of about 2 cm/sec it would require about 6 years to spread
over the area between 45''N to 15° N. During this time the Straits of Gibraltar will
supply 312-000 km^ of Mediterranean Water, which, distributed evenly over a layer
of 500 m thickness from 45° N to 15° N., would mean a contribution of about 3-4%.
The layers inside the Spanish bay will, of course, show a considerably higher percen-
tage.*

IsELiN (1936) has not quite agreed with the idea of an extension of Mediterranean Water to the
higher latitudes of the Southern Hemisphere. On the basis of "Atlantis" observations he investigated
the deviations of individual values from the standard value for the whole region using the Helland-
Hansen anomaly method (see Pt. I, p. 114). A positive anomaly is present at 1200 m depth only as far
as about 20° N. (until the North Equatorial Current), while farther south deficits appear due to the
effect of mixing with Antarctic intermediate water. According to Iselin the effect thus extends no
further than 20° N. This difference in viewpoint can be explained by differences in the definition of
the "Mediterranean Water"; the fact at least remains that traces of Mediterranean Water can be
followed far into the South Atlantic.

The process of spread of Mediterranean Water through the Straits of Gibraltar
and out into the Atlantic is certainly of a twofold nature. During the first part of the
outflow and sinking of the heavier Mediten-anean Water, until it reaches the shelf and
the continental slope and until it finds the depth of equal density inside the Atlantic,

* These percentages refer to the water present between 600 and 700 m depth west of the Straits
of Gibraltar which has a temperature of 11-9° C and a salinity of 36-5 %„ and was termed "Mediter-
ranean Water" by Wiist. If absolute values are required of the proportion of Mediterranean Water
from east of the Straits of Gibraltar then the given values must be reduced by half.

The Stratosphere Circulation 693

the Mediterranean waters flow with considerable velocity and due to the influence of
the Coriolis force keep especially in the Spanish Bay to the northern side. Finally,
they pass around Cape San Vincent while steadily sinking and still keeping close to
the Portuguese coast past Cape Finisterre as far as the Bay of Biscay. Observations
show that this is the first stage of spreading; and the whole process of spread behaves
exactly in the way described on p. 524 et seq. and in Fig. 251a. The second stage of
spreading starts from this tongue of Mediterranean Water off the Portuguese coast.
Due to the much lower velocities the Coriolis force is no longer effective and the
influence of lateral and vertical mixing becomes dominant. The picture presented
in Fig. 330 is thus an effect of mixing processes and Defant (1957) has shown that
a lateral eddy viscosity coefficient of about 5-5 x 10'' cm^ sec~^ is quite sufficient to
explain the lateral spread. A precise account of the whole process, however, requires
systematic series observations and current measurements along suitable sections.

The Indian Ocean also shows in all meridional salinity sections, starting from the
Gulf of Aden in the north-west, an unmistakable effect of the highly saline waters
spreading out from the Red Sea through the Gulf of Aden into the Indian Ocean. In
this case also the effect of this outflow is of decisive importance for the stratospheric
circulation.

It is only to be expected that there will be seasonal variations in the extent of the
spread of the water from the subtropical adjacent seas, since the outflow in itself is
known to be subject to rather strong variations of this period (p. 503). The observa-
tional data available at the present time do not allow to show the influence of such
seasonal fluctuations in the open ocean.

Investigations of the extent of spread of the Mediterranean Water show the great
importance of the subtropical adjacent seas for the deep-sea circulation. Due to the
high density of the water masses flowing with the deeper currents into the open ocean
the layers of the stratosphere will have a sinking tendency and form a source for the
onset of large-scale circulations. This source is at least as important as the convection
acting from the sea surface downward in polar and subpolar seas. These inflows are
also important because of another reason. Mixing of the water masses transported
by these currents with tropospheric water masses above causes an interaction between
the oceanic troposphere and stratosphere, and direct exchange between the two main
layers of the ocean is probably restricted to these places. While the Atlantic and Indian
Oceans are affected by subtropical adjacent seas, the outflow from which considerably
intensifies the circulation in the uppermost part of the oceanic stratosphere, there are
no adjacent seas of this type connected with the Pacific. Consequently, the Pacific
lacks the large meridional contrast in salinity of the deeper layers which provides the
driving force for a stronger circulation.

Chapter XXI

The Main Features of the General
Oceanic Circulation and Their Physical

Exploration

1. The Oceanic Circulation in the Atlantic

The results obtained by numerous expeditions in the Atlantic allow a complete and,
in itself, closed picture to be built up of the tropospheric and stratospheric oceanic
circulations. Knowledge of the circulation systems in the other oceans is not so precise,
but the conditions in them should not be so very different as is confirmed clearly by
the available observations. An attempt has been made in Fig. 331 to picture the entire
circulation system of the Atlantic in a somewhat schematic meridional section in
order to summarize its main characteristics. This representation applies mainly to
the western side. It can be seen that the main water movements are confined to an
extremely thin layer. The circular representation shows especially the enormous
horizontal extent of the oceanic troposphere. Its vertical thickness is, however, small
so that in spite of the large vertical exaggeration in scale it is difficult to picture the
internal circulation properly in the figure. All the main currents and singular points
of the current system of the sea surface are indicated at the edge of the figure. It should
be remembered that all extensive ocean currents are mainly surface currents and belong
essentially to the oceanic troposphere; they extend down to the water masses of the
oceanic stratosphere in only a few places and to a limited extent. This is especially so in
the tropics and the subtropics.

As compared with the large horizontal extent of the oceanic troposphere the source
regions for the stratospheric water types appear small, nevertheless they remain the
regions of origin for the water movements inside the extended space of the oceanic
stratosphere. In these regions also the forces must be contained for a renewal of the
stratospheric waters and their movements. The effect of the European Mediterranean
which can be regarded as a lateral intrusion from the east appears of no less impor-
tance. The small arrows in the diagram indicate the direction of spread of the individual
water types; the current-like spread is thereby mostly indicated by full arrows while
convectional spread is shown by wavy arrows. The figure shows only the meridional
components of the water movement and deals only with mean conditions. The zonal
components surpass by far the meridional ones especially in the southern part of the
South Atlantic and in middle latitudes in the North Atlantic. The characteristic
asymmetry of the Atlantic circulation and the great importance of the Antarctic for
the stratification and movement of the water masses throughout the entire Atlantic

694

Main Features of General Oceanic Circulation and their Physical Exploration 695

tZ*^

QOO^O''

>o\ dWe'

rqerice

Equatorial counter current
Equotoriol divergence

South

equo/o

Current

i^e/o

'^^-e^

Fig. 33 1. Meridional vertical cross-section from pole to pole through the Atlantic Ocean.
Schematic representation of the tropospheric and stratospheric oceanic circulation.
— ocean bottom, , boundary layer between tropo- and strato-
sphere from Northern polar front to Southern polar front. Salinity distribution:
Fw>n>1 , >36-0%o, B>i?l , 36-0-34-9?4, ^^^ , 34-9-34-6%„, 1^^^ , <34-6%o,
— >-, current-form spreading,-'- ,,', convection-like spreading and convection-like sinking,

exaggeration about 1 :400.

stand out particularly in this diagram. In the north the effects are more sub-arctic due
to the bottom topography, but their influence on the stratospheric water movements is
still extremely important.

If we ask for the driving forces of the stratospheric oceanic circulation it must be
stated that only differences in the thermo-haline structure of the water masses can be
the cause for these circulations, and these contrasts can only be maintained by atmos-
pheric influences affecting the regions north of the oceanic polar fronts and are so
regenerated again and again. Thermodynamic machines of this type can only do work
when the compressions of the medium set into motion occur at a lower pressure
than the expansions (see p. 489 and following pages). The water in the upper circulation
branch is set in motion from a region of smaller to a region of greater density, and in

696 Main Features of General Oceanic Circulation and their Physical Exploration

the opposite direction in the lower branch. The meridional density sections show that
this condition is satisfied and the dynamic evaluation of the observational data has
given proof of the internal forces acting in the pressure field and resulting from the
three-dimensional mass structure.

In the troposphere the thermo-haline circulation in a meridional direction is less
important as compared with the effects of the wind. The air currents therefore set the
characteristic pattern for the circulation here and determine its more zonal direction.
The western and eastern boundaries set by the continents to the oceans, due to the
surface accumulation of water (piling up; Anstau), give rise to gradient currents which
besides the wind drift determine the character of the tropospheric oceanic circulation.

WUst chose a different type of representation to show the oceanic circulation. The
surface currents and the deep-sea circulation of the Atlantic were shown in form of a
block-diagram in order to arrive at a three-dimensional representation and to elucidate
thereby the internal completeness of the circulations (Fig. 332). This survey of the
oceanic circulation teaches that the basic causes of the entire oceanic circulation lie
in the atmosphere. They are due partly to the vv/>7^ which transfers energy to the water,
and partly due to climatic effects on the water masses, especially in polar and subpolar
oceanic regions. These then give rise in the first place to the water movements in the
deep layers.

2. Summary of Present Individual Theories and the Prospects of a Comprehensive
Theory of the General Circulation Including the Deep Layers

The existing theory of the wind-driven circulation in closed oceanic basins has been
found applicable to individual parts of the ocean, but a comprehensive theory of the
wind-driven circulation covering all oceanic parts is so far still missing. It has already
been pointed out (p. 583 et seq.) that the highest advanced theory of Munk and
Carrier (1950, led at least qualitatively to very reasonable results. Criticism has
been expressed primarily on account of the high value of the coefficient of lateral eddy
viscosity required in order to explain the intense currents along western coasts.
Morgan (1956) in attempts to overcome this drawback has examined the necessity
of the inclusion of the lateral eddy viscosity for balancing the wind torque on the
water surface.

The ocean can be represented on a different model from those used previously.
In this it is divided into a northern and a southern part, and attention is paid only to
the southern one which in itself is subdivided into an interior region and a boundary
region adjacent to the western shore. Figure 333 shows these three oceanic subdivisions
and the boundaries between them. The figure contains a typical stream line of the
circulation, most of which or perhaps all of the stream lines can be expected to pass
through all three regions. The equations of motion given for spherical co-ordinates
are formally integrated over the depth both for a homogeneous ocean and for a two-
layered ocean. From these the approximate equations are derived applicable to the
interior region /^ of the currents, that is, to a region sufficiently remote from any
coast. They show that all terms which are non-linear in the velocity components as
well as the terms giving the contributions of the lateral eddy viscosity are negligibly
small there. This is the same result as obtained from the Sverdrup solution. Wind
and Coriolis forces are the principal forces in this region. For the boundary region /,,

Main Features of General Oceanic Circulation and their Physical Exploration 697

3 "a

0) o

o • —

r- C

60 <I

Lu 'njdaa

698 Main Features of General Oceanic Circulation and their Physical Exploration

Fig. 333. The three regions of an ocean model according to Morgan, 1956, Ij, interior
region; lb, frictionless stream region; II, northern region, non-steady, and lateral friction
eflfects possibly in an important way.

an investigation was required to show whether a lateral eddy viscosity is needed to
explain the intense current in the region close to the western shore.

A general discussion based on the momentum balance alone shows that if the
frictional torque is essential to the torque balance it is certainly not the only contribu-
tion to it. On the contrary, a boundary layer analysis, together with an accurate esti-
mate of the order of magnitude of the separate terms, shows the predominance in this
region of the pressure terms, the non-linear inertia terms and the terms arising from
the variation of the Coriolis parameter with latitude. This is in complete agreement
with the theoretical results of Charney (1955) for the Gulf Stream (p. 627) but not
with the result of Munk which presumes here a large lateral friction. In region II,
the non-linear terms, the lateral eddy viscosity and non-stationary effects become of
the greatest importance. Transitions from one region to the other must, of course,
be considered more closely, but it appears that this leads to no further difficulties, so
that it seems possible to obtain a comprehensive picture of the entire ocean circulation.

Very recently Stommel (1957), in an extremely interesting and instructive survey
article, has compared the different theories of ocean currents and discussed their
basic physical ideas. Avoiding mathematical ballast he tried to represent the three-
dimensional oceanic movements by means of schematic block-diagrams, which are,
however, based on strict theoretical principles. It seems not possible to describe all
the details here but only the most essential points in connection with the upper wind-
driven circulation and the deep-sea circulation shall be dealt with. Figure 334 shows a
rudimentary simplified model of the Atlantic Ocean with meridional boundaries 60°
apart in which a certain zonal wind-stress distribution (indicated on the left) generates
a wind-driven circulation. Westerlies prevail between 30° lat. and the poles and the
trade winds extend across the equator from 30° S. to 30° N. The lines shown are
isobars parallelling the geostrophic flow. A certain contribution of the Ekman wind-
driven transport in the surface layers has been omitted in order to retain clarity in the
picture. Obviously, a system of gyres and western currents is obtained as in previous
theoretical investigations. The boundaries between the gyres correspond to the latitudes

Main Features of General Oceanic Circulation and their Physical Exploration 699

of no Ekman layer convergence (see p. 581, Fig. 265); the regions of maximum geo-
strophic meridional flow in each gyre correspond to the latitudes of maximum con-
vergence (meridional flow towards the equator) or of maximum divergence (meridional
flow towards the poles) of the Ekman wind-driven layer. Western boundary currents
corresponding to continuity requirements have been introduced along the western
boundary region.

Fig. 334. The steady circulation produced in an ocean of uniform depth bounded by
meridional coasts 60^ apart, acted upon by a distribution of zonal winds, which are indicated
on the left. The western boundary current is shown schematically by the double line at the
western coast and its transport is indicated by heavy arrows (according to Stommel, 1957).

This simple circulation, derived from the application of previously described
mathematical principles to a homogeneous or a vertically integrated ocean, can now
be interrelated with the internal oceanic circulation of the deep layers which corres-
ponds to a thermo-hahne circulation. For this purpose Stommel subdivided the total
ocean into two layers by means of a level surface half way to the bottom, for instance,
at 1500-2000 m. Across this level surface there is a vertical mass transport which is
specified geographically. The geostrophic flow of the wind-driven circulation super-
imposes on the water transports of the internal circulation and the continuity condi-
tions require that the vertically integrated transport over both layers together should
vanish. Figure 336 shows this model given by Stommel, a similar kind of presentation as
used for the previous model. A level surface L divides the ocean into an upper and a
lower layer. The thermo-haline convection processes allow a sinking of the water
masses across the level surface in sub-Arctic latitudes (p. 684) and a corresponding
rise across the level surface in sub-Antarctic latitudes (p. 675).

These convection processes are indicated by vertical transport lines drawn through
the level surface in Fig. 335. The remainder of the thermo-haline circulation is com-
pletely determined by continuity and dynamic reasoning; the transport of water
between the two hemispheres takes place in a narrow western boundary current
according to the dynamic principles frequently mentioned above that are effective
on the rotating earth. The field of motion in this circulation is entirely internal; its

700 Main Features of General Oceanic Circulation and their Physical Exploration

vertically integrated transport vanishes at all points. This internal thermo-haline
circulation postulated by Stommel is in full accord with the deep-sea currents deduced
from the observations of the "Meteor expedition. The sinking of sub-Arctic water
masses in the Iceland-Greenland region (p. 684), the concentration of the North
Atlantic Deep Current close along the western side (p. 672), the rise in the sub-
Antarctic region of the water masses carried southwards (p. 687) and the sub- Antarctic
intermediate current flowing north (p. 679) are the principal constituents of this
internal thermo-haline circulation which derives its driving force from the density
differences between the sub-Arctic and the sub-Antarctic oceanic regions. The defi-
ciency of the Stommel representation of this internal circulation is that in the Atlantic
as in the other oceans, the Antarctic Bottom Current in which the Antarctic water after
sinking at the continental shelf into the deepest troughs flows northwards beneath the
sub-Arctic branch of the thermo-haline internal circulation (lower Atlantic Deep
Current), penetrates further into the North American Basin and after mixing with the
upper waters is carried south again in this current (see Figs. 323 and 331 ).

In Fig. 335b, Stommel now shows separately a wind-driven circulation in the upper
layer corresponding to Fig. 334, except for the additional gyre just north of the equator
caused by the presence of an area of doldrums in the wind field there (p. 601). Both

Fig. 335. A schematic interpretation of the circulation in the Atlantic Ocean constructed by
a superposition of an internal thermo-haline mode associated with a flow across a level
surface L at mid-depth (a) and a purely wind-driven circulation in the surface layers (b).

The sum of these two is shown in Fig. (c). According to Stommel, 1957.

Dashed arrows indicate portions of flow not given by elementary theory but evidently

required by continuity and sketched in.

Main Features of General Oceanic Circulation and their Physical Exploration 701

these circulations are now superimposed on each other in Fig. 335f. Particularly
noticeable is the absence of the Brazil Current and the intensity of the Gulf Stream,
even though the vertical integrated transport is the same for both currents; but
according to the interpretation suggested by Stommel the current in the deep layers
opposes the Gulf Stream but flows in the same direction as the Brazil Current. The
Gulf Stream is reinforced by the thermo-haline component but the Brazil Current is
so weakened that it almost disappears. This picture of the circulation of the Atlantic
Ocean is undoubtedly interesting and instructive and will stimulate further thinking
and conclusions which, however, must be supplemented by corresponding further
oceanographic surveys and current measurements in the deeper layers of the oceans.

3. Model Experiments on Stationary Planetary Flow Patterns

Thoughts about the physical fundaments of the oceanic circulation lead to an
analysis of simple flow patterns in a homogeneous fluid layer: (1) of uniform depth on
a rotating sphere and bounded by meridional barriers; (2) of uniform depth on a
^-plane (plane with j8=2a» sim ^ — const, see p. 556) and bounded by barriers running
north-south; (3) of radially non-uniform depth on a rotating plane and bounded by
radial barriers. Analyses of this type and associated model experiments have been
made recently in a very instructive form by Stommel, Arons and Faller (1958).
Although these investigations cannot be regarded as concluded they, nevertheless,
throw some light on the physical processes operating in the oceanic circulation, so
that it seems appropriate to present the main contents of these investigations here.
The essential elements which define the simplified regime described above are:

(a) the flow in the whole layer is steady and geostrophic except and only;

(b) at the western boundary where a narrow intense western boundary current is
permitted to depart markedly from geostrophic conditions, and moreover;

(c) this system which would otherwise be at rest is driven by a distribution of fluid
sources and sinks representing various diff'erent driving agents such as the wind. This
is no real restriction.

Some of these analyses and thoughts were tested by experiments in a pie-shaped
sector of a fluid basin rotating counter-clockwise. The free surface was a paraboloid
cylinder with vertical axis, concave upwards. The undisturbed depth varied radially
from a minimum at the centre to a maximum at the outer rim. The top was covered
with a sheet of glass in order to prevent the air in the room exerting any stress on the
surface. After rotation for some time the fluid is completely at rest relative to the basin.
There will be a component of flow radially outward (or inward) in the interior of the
fluid only if there is a local fluid source (or sink). Components of geostrophic flow
along circles of constant radius are permissible without divergence except where
blocked by radial barriers. Figure 336 shows possible variations in the relative distribu-
tion of sources and sinks and the currents that would be expected in each case. The
point source @ and sink 0 of equal intensity were placed :

(a) near to the eastern boundary of the sector at the end of radii of diff'erent length;

(b) as an isolated source only at the apex of the sector. Since no point sink is pro-
vided the free surface will rise uniformly;

(c ) as an isolated source at the western edge of the rim.

The current flows along the shortest path to the western boundary of the sector.

702 Main Features of General Oceanic Circulation and their Physical Exploration

Fig. 336. (a) Diagram of circulation induced in a rotating sector by a source ® and a
sink 0 positioned as shown, (b) Sketch of flow pattern expected with source @ at apex of
sector, surface of fluid rising uniformly, (c) Sketch of flow pattern expected with source ©
at western edge of rim, surface of fluid rising uniformly (according to Stommel, Arons and

Faller, 1958).

There a narrow intense western boundary current develops always, and in (a) after
reaching the outer rim the water returns in another zonal geostrophic current to the
sink 0. In (h) there develops a narrow intense western boundary current and in a
surprising way the basin fills up from the rim although water is added at the apex.
Even more surprising is case (c). The sector is allowed to fill up from the isolated source
at the western edge of the rim. The interior geostrophic flow is again directed towards
the centre, but the interior radial transport is so large that it feeds at the apex a narrow
western boundary current which flows back towards the source ©.

The theory of these processes explains convincingly the nature of the water transports
and explains the formation of the western boundary current which governs the process,
though without, however, giving any detailed dynamic explanation. To check and to
illustrate the principles of the theory and quantitative ideas concerning the flow in the
rotating sector, Stommel, Arons and Faller have made rotational experiments in a
tank with the form of a truncated sector of 60° width. In Fig. 337 are shown the
experiments corresponding to those of Figures 336a and c; these confirm clearly the
qualitative and theoretical argument.

The application of the results of such experiments to phenomena which can be
observed in the ocean, is readily understood and their further development with the
guidance of carefully chosen theoretical models should contribute much to an under-
standing of the phenomena occurring in ocean currents.

4. The Transient Response of an Ocean to a Variable Wind Stress

In all theoretical investigations of the ocean circulation induced by zonal winds it
has been assumed that the effect of the wind does not change with time (is constant
with time). It is known, however, that this is true only for a first approximation and
attempts have occasionally been made to study the effect of a wind that changes with
time on vertical structure and circulation of an ocean. A study of the time-dependent
wind-driven circulation in a homogeneous, rectangular ocean has been given by
Veronis and Morgan (1955). Already somewhat earlier the problem has also been
considered by Ichye (1951). They start essentially from the same equations of motion

Fig. 337. (a) a photograph at 20, 80 and 220 min after the introduction of dye. The source
was at the apex and there was no external sink (corresponding Fig. 336/?). (h) Photographs
at 5. 10 and 20 min. The dyed fluid was injected through a vertical glass tube in the south-
west corner of the tank and the sink was at the apex (corresponding Fig. 33('i(). (According
to Stommel, Arons and Faller, 1958.)

Main Features of General Oceanic Circulation and their Physical Exploration 703

as used by Munk (1950) and Hidaka (1950) taking into account the lateral eddy
viscosity and obtain, for a zonal wind stress whose amplitude varies harmonically
with time, the variations in the strength of the currents and the phase lag behind the
wind in the individual circulation gyres. The results seem to be somewhat outdated by
the more recent developments in the theory of the general oceanic circulation.

A new and rather important contribution to the effect of a time-dependent wind
on a stratified ocean has been made by Veronis and Stommel (1956). Now one deals
with non-stationary conditions, which stand in question and which can in general
be regarded as aperiodic disturbances across the given current field ; these disturbances
of rather different dimensions may therefore vary with both time and position. A model
was used in which the ocean was taken as horizontally unlimited — coastal effects
were thus disregarded — and it consists of two layers (an upper and a lower layer
separated by a boundary surface). The wind system introduced, however, is of a finite
size. In agreement with the theoretical work on the dynamics of ocean currents in the
central parts of the oceans (Sverdrup, 1947 and Reid, 1948) the lateral eddy viscosity
was disregarded. The theoretical investigation tends towards an understanding of the
way in which a two-layered ocean would react to changes in the wind field acting on it.
The main questions were as follows :

(a) will the wind-generated current restrict itself to the top layers so that the hori-
zontal pressure gradients and the velocities in the deep layers could be neglected, and
how will the boundary surface and the physical sea level behave under these conditions ? ;
and

{b) will the wind-driven current extend down to both layers, and is the horizontal
pressure gradient in both layers down to the sea bottom of the same order of magni-
tude?; or

Movements of type {a) are called internal or baroclinic, those of type {b) external
or barotropic. This is illustrated by the scheme given in Fig. 338. It has often been

Approximatly ^
ctrest, ^~0

Type (a): baroclinic

Type (0 : barotropic

Fig. 338. The type of motion in a two-layered ocean, (a) baroclinic or internal type, with

a motion in the upper layer and a nearly motionless lower layer, (b) barotropic or external

type. Horizontal pressure gradient nearly equal in both layers.

704 Main Features of General Oceanic Circulation and their Physical Exploration

pointed out in previous work that for a time-variable wind effect the three-dimensional
structure of the sea tends to be baroclinic, which is in fact fairly readily understood. If
the physical sea level inclines due to a wind influence the correspondingly generated
horizontal pressure gradient at first extends down to the bottom also in the case of a
two-layered ocean. In addition to the flow in the uppermost layers, more or less in
the direction of the wind producing it a flow in the lower layers will thus also occur,
only in the opposite direction (see Fig. 339). This latter current will increase in intensity
until the slope of the internal boundary surface (opposite to that of the sea surface)
causes the horizontal pressure gradient to disappear in the lower layers, so that the

Wind

■--^loce;^

^Oc^.___.

p decreosinq

p increasing

mmmimyM

zmmm^M

Fig. 339. Transition from a barotropic type in the first phase of wind influence to the final

baroclinic type.

lower part of the ocean will finally be at rest. In the upper layer there will then be a
drift current and underneath a geostrophic flow, while in the lower layers the ocean is
at rest. It is therefore to be expected that the mass field of the sea and the wind-
generated currents of the upper layers act and react on one another and this inter-
relation is such as to restrict the wind-driven currents to the upper layers of the ocean.
This striking compensation principle between the upper and lower layers is confirmed
by experience and is one of the most important experimental facts of oceanography.
If this were not the case it would not be possible to build up a picture of oceanographic
conditions in the deep layers and their mass displacements on the basis of wide-spaced
oceanographic observations; that is, it would be impossible from observations at
widely differing times to form a picture of the average conditions of stratification and
field of flow in the deep layers. This supports the theoretical results, since if these
were somewhat different there would undoubtedly be a contradiction with experience
and the model chosen would be unsuitable for such a purpose. This problem was
first discussed by Rossby (1938), but his results were unsatisfactory since they gave
more or less barotropic flow systems which is impossible. A barotropic state can only
persist for a short time and finally a baroclinic state must predominantly prevail.
After other attempts Veronis and Stommer'ihave re-examined the problem and
attempted its solution by means of a large mathematical apparatus.

Main Features of General Oceanic Circulation and their Physical Exploration 705

In the two-layered oceanic model there are as usual two equations of motion for
each layer, one for the w-component and the other for the f -component of the velocity,
and the continuity equation. The equations of motion for the upper layer therefore
take into account the wind stress acting on the sea surface. This then gives three pairs
(2 times 3) of differential equations. By cross-wise differentiation, and taking into
account the variation of the Coriolis parameter with latitude, one obtains for each
layer a vorticity equation. As a first assumption the movements are taken as indepen-
dent of the >'-direction. As a consequence, the problem is thus one-dimensional and
the equations are considerably simplified. This gives two equations which permit
a study of the reponse of the physical sea level and the internal boundary surface to
the variable shearing stress of the wind. It is interesting from the mathematical point
of view that these equations can be combined to give an equation with a single variable
without raising the order. It is of the fourth order and has the form

1 R T'

A k Rxxxt 72 ^xttt P^xt "T HA k Rxx 7^ ^tt ^= ~7~ • (XXI. 1)

R has a fixed numerical relationship to the displacement of the sea surface and the
internal boundary surface and can have two values, 7?^ and i?2- In the same way k has
the numerical values ki and k^ corresponding to the values R^ and Ro. Moreover,
^^ (= gDJf^) where Dg is the equilibrium thickness of the lower layer and A is a
quantity termed by Rossby the "deformation radius". The solution of the differential
equation (XXII. 1) gives the "normal values of motion" (equation of normal modes)
and makes it possible to determine all the desired quantities of the model such as the
displacement of the boundary surfaces and the velocity in the different layers.

This equation can be used to derive the free waves of the system and their depen-
dence on the dimensions of the system, when the wind stress is omitted in the equation.
A knowledge of the free waves is of considerable value because of its great importance,
since in view of resonance phenomena they may have considerable influence on the
forced waves which are generated by the action of the wind. Assuming a normal mode
of the form

Ri = Si sin (Ix + ojit ) (/ = 1 , 2) ; (XXI.2)

that is, in form of a wave progressing in the negative x-direction with a frequency
coi, then the equation (XXII. 1) transforms into

(/)'

fl\f

with

[j-^ +(l+^.)_!+^ = 0 (XXI.3)

The solution of this algebraic equation (three positive roots) gives the frequency <u,-
as a function of the wave-number l-nll or of the wavelength L. The roots are a»,-,i;
^i,i\ ^1,3 and correspondingly there exist in total six possible modes of wave motion.
Figure 341 gives frequencies and periods of the waves for D^ = 500 km, D^ = 3500 km,
/= 10-* sec-\ /S ^ 2 X 10-"m-i sec-^ and for wavelengths 10 < L < 12,000 km
covering the entire region under consideration.

706 Main Features of General Oceanic Circulation and their Physical Exploration

Two of the waves have large periods and in these the flow is in geostrophic equili-
brium; they are the barotropic and baroclinic Rossby waves. The other four waves,
two of which are barotropic and the other two baroclinic, are inertial-gravitational
waves resulting from an imperfect balance between the pressure and the Coriolis force.
In general, the barotropic waves are pure gravitational waves with a velocity of propa-
gation \/{g{D-i^ + D^)} the baroclinic waves are pure inertial waves with a period of a

WAVE NUMBER 5 CM"'

WAVE LENGTH, KM

Fig. 340. The velocity and frequency of all the various free waves, which may occur in a
two-layered ocean (according to Veronis and Stommel, 1956).

half a pendulum day 2ttJ /; at the short branch in their connection on the right (see
Fig. 340) are ordinary internal waves at the boundary surface with periods of between
1 h and 1 day.

This derivation of all possible wave types from a single equation is extremely interes-
ting and instructive. Two types of disturbances in time will be taken for a study of
wind-driven motions. In the first case they are forced waves generated by a moving
wave-wind system. This wind system as to the order of magnitude shall be comparable
with atmospheric disturbances as are shown in 5-day average charts. This corresponds

Main Features of General Oceanic Circulation and their Physical Exploration IQTl

to a period of about 2 weeks and a wavelength of about 6000 km. The force producing
them thus has the form fFsin (Ix + vt ), where W is about 1 cm^ sec"^; for an east-
wards movement of the disturbance v is negative. For periods of 1-7 weeks — values
which are comparable with the periods of barotropic Rossby waves — the ocean
reacts largely as a homogeneous water body. As the period increases the baroclinic
effects become also larger and for longer periods (more than a year) the motion is
only partly barotropic and the baroclinic effects will be more important. For very long
wind-periods (at least about 100 years) the motion is entirely baroclinic. The flow is
geostrophic and in full accord with a stationary state.

The second type of wind-driven ocean currents is that produced by a stationary
wind field imposed suddently at a given time. In this case all the possible free waves of
the system may develop and an investigation can be made of the relative importance
of inertial-gravitational waves and of geostrophically balanced motions.

If the action of the wind lasts for a period comparable with that of an ordinary storm
then the geostrophically balanced motion will be partly barotropic and partly baro-
clinic. The internal boundary surface also reacts on the wind influence and this effect
can definitely be found (10-20 m), if the wind continues for 3 or more days. The deep
currents, however, remain weak and are probably no stronger than the thermo-haline
currents such as those produced by Antarctic cooling. The effect of storms can thus
make little contribution to the large-scale lateral mixing inside the oceanic stratosphere.

Other movements of inertia and gravitational character which may be generated
are stronger but are not accompanied by measurable displacements of the boundary
surface ; they are pure horizontal inertia oscillations without any horizontal pressure
gradients and depend largely on the earth rotation.

These investigations of Veronis and Stommel are undoubtedly of great import-
ance for a knowledge of the dynamics of the ocean currents. They are, of course, so far
incomplete; they do not, for instance, provide an explanation for the effects of
barriers (coasts and the sea bottom) as well as for the effects of friction. At the
present time, however, it is sufficient to gain some insight into the time-variable
action of the wind. This is all the more important because of the extreme difficulty of
gaining an insight into such rapidly changing phenomena solely by means of oceano-
graphic observations.

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Author Index

Prince Albert I of Monaco, 342
AiTKiNS, W. R. G., 55, 57
Albrecht, F., 236
Anderson, E. D., 116, 122
Angstrom, A., 59, 91, 92, 224, 225
An vers, H. G., 8, 624
Arons, 701, 702
Arx, von W. S., 617. 634
aschkinas, 51

Bein, W., 43, 57

Benard, H., 199

Bergeron, T., 300

Bergten, E., 8

Bernoulli, 325. 326, 328, 374

BiGELOW, B, 593

Bjerknes, J., 300, 453, 469

Bjerknes, v., 42. 300, 303, 306, 308, 313, 315,

329. 357, 359. 365, 366, 372, 375, 434, 451, 466,

476, 486, 491, 577
Bohnecke, G., 32, 35, 111, 112, 140, 141, 154,

161, 186, 190, 566, 663
Bowen. J. S., 225
Bowie, W., 8
Bozin, E., 644
Breitfuss, L. L., 137
Brennecke, W., 12, 73, 74, 149, 341, 418, 422,

437
Brockamp, R., 256
Brooks, C. E. P., 264, 559
Bruch, H., 221
Bruckner, E., 232, 234
BucH, K., 71, 72, 74, 76, 77, 80, 83
BUCHAN, A., 561
Buchanan, J. Y., 43, 575
Budel, J., 257, 263
Buen de, R., 182, 529
Bullard, G., 12, 89
BuMPus, D. F., 634
Bundgaard, R. C, 300

Carnot, 490

Carpenter, 575

Carrier, G. F., 585, 587, 696

Carrit, 65

Carruthers, J. H., 17, 342, 343

Cassini, 287

Castens, G., 478, 493

Charney, I. G., 627, 629, 630, 631, 698

Cherubim, R., 222

Chevalier, 48

Church, Ph. E., 143, 144

Clarke, G. L., 55

Clowes, A. J., 148, 672

COLDING, A., 420

Collins, 51, 52

Cooper, L. H. V., 22, 680

Crary, a. p.. 12
Croll J., 575
Cromwell, T., 604

Dall, W., 641

Dallas, W. L., 666

Daly, R. A., 22

Deacon G. E. R., 72, 144, 145, 148, 149, 173,
505, 550, 666, 669, 670

Defant, a., XV, 1, II, 27. 28, 95, 98, 104, 105,
106, 107, 109, 115, 121, 122, 136, 138, 144, 148,
153, 160, 164, 166, 173,204,211,232,283,317,
344, 345, 346, 352, 376 378, 387, 392, 305, 413,
435, 449, 453, 458. 463, 468, 473, 476, 480, 494.
498.500. 501, 517. 519, 535, 556, 565, 583, 593,
596, 598, 620, 644, 646. 649, 652, 660, 662,
663, 678, 679, 680, 682, 687, 693

Defant, Fr.. 442, 443, 444, 445. 446

Dietrich. G., 5, 8, 31, 42, 51, 56, 109, 116, 117,
132. 133, 134, 236, 460, 493, 494, 501, 505, 507,
559, 606. 607, 623, 624, 641, 642, 643, 680

DiNKLAGE, L. E., 417, 418

Dittmar, C, 37, 38, 74

DORN, VAN W. G., 421

Drygalski, E. v., 244, 260, 261, 273, 274, 275

Durst, C. Sc, 418

Ekman, F. L., 380

Ekman, V. W., 34, 42, 49, 104, 125, 195, 315,
387, 391, 399, 401, 403, 404, 406, 408, 409, 412,
413, 414, 417, 420, 422, 426, 427, 428, 431, 443,
444, 446, 447, 449. 480, 482, 484, 485, 493, 503,
508, 509, 510, 540, 541, 548, 549, 552, 572, 576,
581, 582, 620, 623, 624, 647, 699

Emden, 95

Emmel, V. M., 65

Ericson, D. B., 22

Ertel, H., 103, 160, 232, 404, 477

Euch, 83

EuLER, L., 320, 383

Ewing, M., 12, 22, 256

Exner, F., 418, 443, 465, 469

Falkenberg, G., 60

Faller. 701, 702

Felber, O. H., 565

Ferrel, W., 575

Fischer, K., 232

Fjeldstad, J. E., 104. 115, 404, 443, 557

Fleming, R. H., 51, 105, 107, 644

Forthman, E., 620

Foronoff, N. p., 604

Forch, C, 416

Fox, Ch., 65

Fritsche, E., 249

721

722

Author Index

fuglister, f. c, 561
Fulton, T. W., 342

Galle. p. H., 416
Cans, R., 61
Gedge, H. D., 532
Gehrke, J., 101, 159, 381
Geissler, H., 36
GODSKE, C. L., 300

GOERTLER, H., 434

GoLDSBROUGH, G. R., 573, 574
goldschmidt, v. m., 81
Goldstein, 392
Granquist, G., 159
Gray, 256

Groen, p., 504, 552, 553, 554
GULDBERG 398, 399
Gunther, E. R., 571, 645
GUSTAFSON, T., 392,

Haber, F., 40

Hadamard, J., 554

Hahn, a., 8

Hamberg, a., 74, 248

Hann, v. J., 117

Hansen, H. E., 267

Hansen, W., 588, 638

Hanzawa, M., 396

Harvey, 65

Haurwitz, B., 637, 659

Hecson, 22

Hegemann, F., 266

Heiskanen, 301

Heissenberg, Th., 396, 397, 398

Hela, J., 420, 449

Helland, a., 273

Helland-Hansen, Bj., xv., 41, 46, 114, 125, 189,

199, 202, 346, 368, 435, 446, 447, 478, 486, 493,

502, 503, 505, 506. 563, 659
Hellstrom, B., 547
Helmholtz, v. H., 453
Hentschel, E., 563
Hepvorth, C, 559
Hess, H. H., 26
Hesselberg, Th., 42, 195, 309
Hessen, K., 7
Hidaka, K., 36, 377, 444, 470, 588, 652, 653, 673,

703
Hildebrandsson, H. H., 283
Hjort, J., 531
Hirsekorn, H. G., 43

HOMEN, 109

Hoover, 601

Humboldt, v. A., 1 19, 575

Huntsman, A. G., 264, 265

ICHiYE, T., 640, 702
IDRAC, p., 532

Jacobs, W. C, 225, 236, 242
Jacobsen, J. p., 101, 102, 104, 170, 207, 208, 210,
211, 212, 214, 392, 502, 503, 527, 529, 537, 607
Jaenicke, J., 40
Jakhelin, a., 510
Jeffreys, H., 200, 383, 404, 492

Jensen, 502

Jerlov, N. G., 51, 56, 605
Jessen, O., 22
Jnoue, 396
Joseph, J., 54, 56

Jntern. Hydrographic Bureau, Monaco, 4
JsAACS, J. D., 333

Jselin, C. O d., 333, 493, 494, 505, 563, 607, 613,
614, 629, 692

Kaehne, K., 22

Kalle, K., 36, 39, 41, 60, 61, 63, 64

KiiLERiCH, A. B., 669

KisiNDO, G., 638

Kleinschmidt, E., 224

Knudsen, M., 34, 37, 39, 73, 379, 381, 527, 529,

534
KoENUMA, K., 115. 638. 640
KOEPPEN, W., 641
KOLMOGOROFF, 396

KossiNNA, E., 2, 3, 5, 14, 15, 17, 18, 21

Kossmat, F., 24

Kreusler, 51, 52

Kruger, W., 343

Krummel, O., 8. 43, 50, 51, 141, 153, 261, 342,

372, 382, 547
KuENEN, Ph. H., 22
Kuhlbrodt, E., 92, 110
Kullenberg, B., 22, 392, 447, 448, 449

Lagrange, de J. L., 320. 322. 324, 342

Lambert, W. D., 301

Landolt-Bornstein, 5, 42

Laurila, E., 420, 544

Lauscher, F., 59, 60. 61, 91

LE Danois, E., 659, 660

Lenz, E., 575, 576

Lettau, H., 81, 82, 236

Leverkink, 8

Lutgens, R., 222

Lumby, J. R., 34

McEwen, G. F., 104, 645, 646

Mae, H., 533

Makaroff, 247, 256

Makin, C. J. S., 38

Malmgren, F., 117, 245, 246, 247, 248, 249, 250,

251, 253, 254, 255
Mao, 601

Margules, M., 453, 456. 474, 635
Marmer, H. a., 8
Marvin, C. F., 221
Matthews, D. J., 41
Maustrad, a., 244
Maurer, H., 301
Maury, 575
Maxwell, A. E.. 89
Mecking, L., 264, 269. 275, 277
Meinardus, W.. 110, 144, 235, 264, 268
Menendez, N., 532
Mercanton, 273
Merz, A., 109, 122, 203, 357, 367. 368, 373, 390,

516, 523, 567. 672
Meyer, H. H. F.. 32, 39, 357. 369, 370
Michaelis. G., 357, 567
MiEGHEM, van J. M., 635
Model, F., 421

Author Index

723

MoHN, H., 220, 252, 398, 399, 418, 476, 493
MoLLER, L., 43, 148, 172, 373, 380, 390, 514, 516

517, 523, 524, 564
Montgomery, R. B., 170, 193, 195, 196, 222,

229, 231, 412, 413, 598, 604, 605
Morgan, G. W., 696, 698, 702
MosBY, H., 36, 91, 224, 225, 390, 505
MosBY, O., 665. 667, 668, 681
MOSKATOV, 257
MOSSMANN, R., 234
MOTHES, H, 256
MuNK, W. H., 116, 122, 231, 421, 422, 583, 584,

585, 586, 587, 588, 589, 631, 673, 674, 696, 703
Murphy, R. C, 572
Murray, J., 531

Nansen, Fr., 43, 67, 97, 123, 189, 346, 354, 355,

368, 399, 418, 476, 493, 505, 563
Nares, G. N., 531
Nernst, E., 48, 243
Neumann, G., 68, 69, 116, 134, 156, 158, 159,

187, 200, 421, 426, 497, 498, 561, 562, 589, 590,

626
Neumayer, v. G., 557
NiKiTiN, W. N., 68, 134
NoMiTSU, T., 8, 405, 444, 516
Nusser, F., 263

Okada, M., 377, 378
Okamoto, G., 8, 569
Omi, 421

OSTER, 55

Otterstedt, B., 447

Paech, H., 567

Palm^n, E., 356, 417, 418, 419, 420, 440, 544

545, 546, 547, 598, 605, 673, 674
Panofsky, 637

Parr, A. E., 105, 132, 192, 505, 507, 606, 607, 624
Penck, a., 8, 20, 220
Perlewitz, 12
Pernter, J. M., 54
Pettersson, H., 12, 40, 43, 54, 391
Petterson, O., 43, 97, 116, 247, 249, 250, 281
Pillsbury, 607
POLLAK, M. J., 197
Poole, H. H., 55, 57
Prandtl, L., 175, 373, 388, 389, 411, 412, 421,

620
Pratje, O., 11
Priebsch, J., 619
Proudman, J., 395
Puff, A., 567
PuLS, C, 569

QUENNEL, W. A., 264
Query ain, de A., 273

Rakestraw, N. W., 65, 606
Ramanathan, K. R., 61
Rankama, K., 82
Rappleye, H. S., 624
Rauschelbach, H., 348, 362
Ravenstein, p. R., 9

3A*

Rayleigh, Lord, 200

Reichel, E., 236

Reid, R. O., 542, 551, 553. 599, 601, 602, 603

Revelle, R., 51, 89

Reynolds, O., 328, 393

Richardson, L. F., 392, 394, 395, 396, 634

RiEL, vanP. M., 127, 130, 131

Rietschel, E., 8

Riis Carstensen, E., 275

Ringer, W. E., 247, 250

RoMER, E., 362

Rossby, C. G., 335, 412, 413, 421, 494, 583, 617,

619, 620, 621, 622, 623, 624, 625, 631, 632, 640,

658, 704, 707
RoYEN, N., 256
Ruden, p., 175, 620
RuppiN, E., 12, 50
Rutherford, H. M., 12
ruttner, p., 54

Sahama, T. G., 82

Sandstrom, J. W., 42, 365, 399, 469, 486, 489,
547

Sarasin, E., 56, 57

Sauberer, F., 54

Sawyer, 51, 52

Schmidt, W., 53, 54, 59, 90, 102, 104, 110, 128
221, 223, 224, 391, 423, 425, 426

Schokalski, J. M., 68

Schott, G., 60, 111, 127, 140, 148, 161, 170, 171,
181, 182, 188, 235, 269, 370, 436, 493, 529, 531,
537, 538, 557, 567, 569, 571, 572, 600 645 687

Schubert, v. O., 127, 197, 198, 345, 660

SCHULZ, B., 70, 72, 74, 127, 136, 266, 343, 563

Schumacher, A., 34, 36, 41, 357, 359, 362, 370,
436, 559, 560, 561, 644

Seiwell, H. R., 67, 104, 508, 606

Shaw, W. N., 559

Shepard, F. p., 21, 22, 421

Shouleikin, W., 222

Skorzow, 134

Sigematu, R., 24, 638

Skogsberg, Tage., 115

Simpson, 139

Smith, E. H., 259, 270, 272, 274, 275, 276, 279,
281, 459, 488, 505, 606, 665, 667, 668, 681

Smith, P. A., 21, 22, 27

Soret, J., 56

SouLE, F. M., 505, 665, 667, 668, 681

Speerschneider, C. J. H., 269

Spilhaus, a. F., 36, 143, 601

Staff, 104

Stahlberg, W., 11

Stefan, J., 252

Stenius, S., 48

Stockman, W. B., 104

Stocks, Th., 12, 17, 19, 27, 30, 31

Stommel, H., 105, 200, 496, 499, 500, 581, 582,
583, 584, 587, 589, 591, 627, 631, 634, 635, 674,
676, 698, 699, 700, 701, 702, 703, 704, 706, 707

Stroup, E. D., 604

SuDA, K., 104

SuESS, E., 8

SuND, O., 34, 43

SVERDRUP, H. U., 42, 43, 67, 104, 105, 107, 108,
115, 123, 124, 148, 157, 179, 195, 227, 229, 236,
237, 242, 247, 254, 309, 311, 346, 347, 355, 356,
395, 405, 418, 422, 437, 438, 440, 563, 494, 503

724

Author Index

SvERDRUP, H. U. — contd.

548, 549, 550, 551, 580, 581, 583, 584, 598, 599,
601, 607, 624, 625, 631, 644, 646, 647, 648, 652,
669,671,672,684,688,703

Tait, J. B., 343

Takano, K., 540, 541, 674

Taylor, G. J., 102,317, 392

Thiel, G., 362

Thompson, T. G., 48

Thomsen, Helge, 547, 688

Thorade, H., 41, 104, 107, 343, 346, 347, 348,

349, 350, 378, 405, 418, 422, 428, 476, 511, 563,

571, 598, 624, 645, 646
Thorne, a. M., 256
Thoulet, M. J., 48
Thuras, W., 43
Timonoff, 184
TOLLMEIN, W., 175, 620
Transche, N. a., 260
Tsuchiva, M., 541, 673

Uda, M., 362, 569, 592, 638, 640
Utterback, C. L., 55

Vaux, D., 22, 680
VE^aNG-MEINESZ, F. A., 26
Vercelli, F., 514, 532, 533
Veronis, G., 702, 703, 704, 706, 707
Vine, A., 12
Visser, S. W., 154, 157

Wagner, F., 350
Walker, G. T., 566
Warmer. H., 573

Wattenberg, H., 39, 67, 71, 72, 73, 74, 76, 77, 78,
80, 83, 84, 85, 86, 104, 182, 186, 494, 528, 530,
663, 681, 684, 685

Weenink, M. p. H., 552, 553, 564

Wegemann, G., 110,493

Wegener, A., 7

Weibull, W., 12

Weickmann, L., 140, 270

Weinberg, B., 256

Weiszacker, 396, 397, 398

Wenner, F., 43

Werenskjold, W., 366, 480, 481, 511, 512

Westphal, a., 8

Weyprecht, K., 243

Wheeler, A. S., 38

Whitney, L. V., 59

WiESE, W., 269

WiLLiMZiK, M., 357, 568

WiPPLE, F. J. W., 317

Witting, R., 45, 105, 346, 347, 355, 381, 395, 396,
417, 418, 499, 527

WlTTSTEIN, 61

WiJST, G., 4, 12, 27, 29, 30, 32, 92, 122, 127, 136,
140, 147, 148, 149, 150, 162, 163, 165, 172, 179,
180, 189, 204, 212, 213, 215, 221, 222, 223, 224,
225, 226, 230, 231, 235, 487, 492, 569, 570, 593,
600, 607, 609, 613, 638, 639, 640, 672, 676, 679,
680, 682, 684, 687, 688, 689, 691, 692, 697

WULF, 136

Wyrtki, K., 56

Yoshida, K., 554, 601, 653, 654

Zoeppritz, K., 693

ZORELL, F., 572

ZuBOv (Subov), N. N., 104, 139

Zukriegel, J., 244

Subject Index

Absorption of radiation, see radiation
Adjacent seas, subtropical, effects on deep sea

circulation 690-3
Agulhas current 641-2
Acalinity 72

distribution 73

relation to calcium carbonate content 85

and salinity 74
Antarctic circumpolar current, dynamic of

673-5
Antarctic convergence zone, process in 679-682
Austausch (turbulent exchange coefficient) 92

102, 103
Axis, of contraction 451

of dilatation 451

Baltic Sea, vertical structure of water masses 70

see also North Sea
Barotropy coefficient 308, 341
Benquela current 565
Benard convection cell 199-201
Bjerknes circulation theorem 332

oceanographic applications 486-492
Black Sea, vertical structure of water masses 69
Boiling point of sea water 44
Bosphorus and Dardanelles, current in 5 1 3-526
Bottom polar current 680-3
Bottom water in the oceans 149
Boundary surface between water bodies 451 -469

Calcium carbonate in the sea, as function of
depth changes by chemical and biological
causes 85-7
saturation at surface of Atlantic Ocean 86
solubility 83-5
solution near bottom 86
Canyons 22-4

Carbon dioxide, annual budget on Earth surface
81
dissociation constants 75-7
distribution on surface on South Atlantic 73
exchange with atmosphere 80
in deep places of oceans 77-80
in a section in subtropical part of South

Atlantic 79
partial pressure 71
solubility 71
Chart datum 5, 9

Charts of sea surface currents 557-8
Circulation, oceanic
basic principles 556-561
influence of meridional coast 579 et seq.
mean features in the Atlantic 694
theorem of Bjerknes 330-3

oceanographic applications 486-492
thermo-haline 574-6
tropospheric of tropical and subtropical

oceans 594-604
stratospheric 661-683

theory, of Stommel and Munk 583-591
summary of individual theories 696-8
comprehensive theory 698-701
oceanic and atmospheric-, effects of polar-ice
conditions 279
Colour of the sea 60-4
Compensation currents 370
Computation of velocity of surface currents in
equatorial regions from wind data
552-5
Conductivity, thermal 50, 92, 95, 103
Continuity equation and divergence of current

field 374-9
Convergence, antarctic, process in 669-672
internal structure 671
process at the polar boundary of 656-660
subtropical 144
stream line 359
point 363

theory of disturbance and wave formation
658-9
Convection, autumn and winter, in polar regions
133-140
Benards cell 199-201
dynamic 101
heat exchange between ocean and atmosphere

and 92
horizontal 105
thermo-haline 96-100
Continental slope 16
Conversion of relative in absolute topography

of isobar surface 492-502
Critical discussion on dynamic computation of

oceanographic data 504-8
Current from ships displacement 343
Current measurements,
from a ship 344
correction method 346
difference method 346
smoothing method 347
scientific use of 345-7
elemination of periodic components 351
Current, compensation 370
inertia 441-450
equatorial under 604-5
oceanic, in a homogeneous sea; theory of
382-450

steady, without friction 383
drift 399^06
gradient 406-13
elementar 413-9
effects of coast on 426-8
oceanic effects, of bottom topography
428^30

of friction 432-6
of varying latitude 420-4
in a non-homogeneous sea 474-512
and density field in a horizontal plane
476-9

725

726

Subject Index

Current — contd.
caused by excess of precipitation and run-off

over evaporation 562-4
density, effect of wind on 544-555
relationship between wind and 550-2
surface density, computation of velocity in
equatorial regions from wind data
552-5
steady in a stratified ocean 479

including friction 482
stationary, and water bodies 451-469
system in a hydrographic vortex 578-9
and thermocline near the equator 463
in sea straits 513-543
theory of 517-523
sea surface currents 557-572
charts of sea surface currents 557
in Atlantic Ocean 558-566
in Indian Ocean 566-8
in Pacific Ocean 568-572
polar currents of the Northern Hemisphere

662-9
antarctic circumpolar currents 673-9
sub-antarctic intermediate 684-9
in the middle part of the stratosphere

683-8
polar bottom 680-3

distributions in the lower Atlantic deep
currents (3000 m) 689

numerical values 103
viscosity 103
numerical values 104
see also viscosity
Effects of subtropical adjacent seas on deep-sea

circulation 690-3
Energy budget between ocean and atmosphere

235-242
Equatorial counter current 559, 569, 599, 602
Equilibrium, condition for static 337
disturbance and re-establishment of static

339
indifferent, stable, labile or unstable 126,

127, 195
quasistatic 338
vertical in the Oceans 195
radiational in uppermost oceanic layers 94
Estuaries (River mouths), current in 538-543

theory of currents in 540-3
Evaluation, dynamic, of oceanographic observa-
tions 338
Evaporation, determination from energy con-
siderations 222-5
distribution 163, 221-5, 229
measurement and computation of 219-21
geophysical process of 225-3 1
Expeditions, oceanographic xiv
Experiments, model, on planetary flow patterns
687-8

Deep-sea bottom 16

data

depressions and trenches 16, 24-7

methods of recording 10

indirect methods with unprotected thermo-
meters 12

large-scale features 27-31

circulation, effects of subtropical adjacent
seas on 690-3
Density of sea water,

dependence on temperature, salinity and
pressure 41

diurnal and annual variations at the surface
185-6

distribution at surface of the oceans 187

potential 192

vertical distribution 191, 194

current, effect on wind on 544-554
Development of oceanography xiii
Diffusion {see eddy diffusivity) 101
Discontinuity surface, stable 453-9
Divergence of stream lines 359

points, 363
Drift bottles and drifting objects 342

current 399-406
according observations 415

East and West Greenland current 662-5
Echo sounding 11

profiles of 19
Eddy
conductivity 92
lateral 93, 105
numerical values 93, 415
diffusivity 103
importance of tongue-like distributions
101, 106

Freezing point of sea-water 44, 45
Friction, Guldberg-Mohn 398

turbulent, see eddy viscosity

velocity 389
Frictional depth and frictional coefficient 422
Fronts, stationary 452

Geoid 6

Gibraltar and Bab el Mandeb, currents in 529-533

Glaciation in the polar regions 271

Glaciers calving into the sea 272

Gliding, up and down-surface 469

Gradient current 406-413

according observations 415
Guiana current 606
Gulf Stream, dynamic of 617-638

comparison with Kuroshio 634-5

internal structure 607-617

main sources of 614

quasi synoptic investigations 615-7
Gulf Stream, stability of 635-7

Charney's theory 627-637

Heat, budget for the ocean 88-90, 223

annual 116

conducted through ocean bottom 89, 128

exchange 88, 92

sources and losses 88, 89, 93
Helland-Hansen's fundamental equation of

dynamic oceanography 486
Hydrogen-ion concentration 74

role in carbon dioxide system 78
Hypsographic curve of the Earth 1 5

Ice, see also Sea-ice
conditions in both polar caps

257

Subject Index

111

Ice — contd.
land, in the sea 271

pack-ice limits in the Antarctic regions 267
pack-ice distribution round of Newfoundland

Banks 265
limits in the Barents Sea for each month 262,

271
limits along the eastern coast of Greenland, in

Davis Strait and Baffin Bay 263
limits in the north-western adjacent seas of

Pacific Ocean 266
character of ice-years around Iceland and in

the Davis Strait 268, 269
polar effect on the atmospheric and oceanic

circulation 279-284
formation in polar regions and autumn and
winter convection 133,137-140
Iceberg, calving, size, shape and destruction
273-5
drift 436-441

in shallow sea 423
in deep sea 424

in the arctic and Antarctic 275-8
productivity in the Arctic 273
south of Newfoundland and of the Grand

Banks 265
seasonal and aperiodic variations in, frequency
off Newfoundland 278
Inertia currents 441-450
in oceanic currents 446-450
periods of oscillating vortex 474
Interchange between sea-surface and atmosphere

235-242
Intermediate subpolar water 173-8, 211-215
Isentropic analysis 192
Isostatic adjustment of the Earth crust 6, 9

Kinematic of the ocean 342

Knudsen's Relations 379

Kuroshio, comparison with Gulf Stream 634-5

internal structure 638-640

surface currents in 569

Labrador current, internal structure 665-9

surface currents in 561
Law of parallel fields 477

Mass field, effect of wind on 544-555
in a limit and stratified sea 544-7
general conditions in the open sea 547-550
Mediterranean, American, current conditions

606-7
Messina, strait of, current 533-4
Mixing length 387-391
processes
lateral 93, 105
vertical 101
Model experiments on planetary flow patterns

701-2
Morphology of sea bottom 12-18
Morphological structure of three oceans 18
Motion of sea level (eustatic, nomic and
juvenile) 8

Neutral point of stream line 364

Nitrogen dissolved, amount in sea-water 66
North Sea and Baltic, water interchange between

526-9
Norwegian Fjords, amount of H2S 69

Oceans, area, volume and mean depth of 1 7

boundaries 1

horizontal extent 1

morphological structure of three oceans 18

three-dimensional structure of 10, 13
Osmotic pressure 44, 47
Oxygen dissolved, solubility 66

consumption 67

vertical distribution 68
Oyashio, surface currents in 569

Peru current, surface 571

Pilling up of water by wind (Windstau) 419

Polar bottom current 680-3

Polar currents of Northern Hemisphere 662-9

Polar front, oceanic 144

Productivity of glaciers calving 272

Production of ice-bergs in the Arctic 273

Radiation, absorption in pure water 52

behaviour of sea-water for diffuse incoming
and outgoing 59

direct solar 90

effective back, radiation in long-waves 59,
91,93

influence of sun's altitude on 90

extinction coefficients 54-6

refraction and reflection 56
Radioactive elements 40
Reference level 492-502

by stations in shallow waters 502-4

determination of 494-501
Refractive index 57
Response, transient, of ocean to variable wind

stress 702-7
River mouth (estuaries), current in 538-562

theory 540-3
Roughness length 390

Salinity, determination 36
periodic variations at surface 1 54
annual variations 156
variations caused by precipitation 159
horizontal distribution at the surface 161
mean meridional distribution 163
vertical distribution 165-6
of oceanic stratosphere 172-3
of homo-haline top layer 166
of subpolar intermediate water 173-8
of the deep water below 1500 m 178
of oceanic troposphere 166-172
in particular depths 179-181
in adjacent seas and sea straits 181-4
Sandstrom's theorem 489-492
Samples, oceanographic 32
Sea bottom 12
topography 18-27

of individual oceans 27-31
and land zones of 5° latitude 3
Sea level 5
mean physical 6, 7

728

Subject Index

Seas adjacent, marginal or mediterranean 4,

30-32
Sea-ice, density and porosity 247-9

formation and terminology 243-5

mechanical properties 255-7

physical and chemical properties 245-257

salinity 245-7

seasonal and aperiodic variations in Arctic
and Antarctic regions 257

thermal properties and temperatures of
249-55
Sea-water, its physical and chemical properties
32-87

principal constituents 37

trace elements in 40

density 41-4

boiling point 44

freezing point 44

optical properties 51

osmotic pressure 44

specific heat 48

vapour pressure of 44

viscosity 50

chemistry of 64-87

oxygen, nitrogen and hydrogen sulphide
contents 65-70

calcium carbonate 83-7

carbon dioxide 41-83
Section, dynamic 338
Shelf, continental 16, 20
Ships journals xiii
Singularities in current field 359-366
Singular lines in current field of Atlantic 564
Specific heat of sea-water 48
Structure, vertical of total Earth 1
Surface tension of sea-water 51
Stability in the oceans 195

in the deep trenches 197

distribution in Atlantic 198
Static of the ocean 337

equation 337
Steady currents in a homogeneous sea

without friction 383

under the action of external forces 398-436

effects of changing depth and spherical shape
of the Earth 385
Straits, sea, water stratification and movements
in 513-7

external influences on oceanographic con-
ditions 534-8
Stratification, stable, of water masses 458-469
Stratosphere, oceanic 122

circulation 661-673

temperature in 123

salinity in 172
Stream line 342

convergence and divergence lines 359

convergence and divergence points 363

neutral points 364

singularities in 359-365
Sub-antarctic intermediate current 675-9
Subtropical convergence 144

Temperature, measurements 34-6
determination in ocean layers 34
changes caused by radiation absorption 94
diurnal variation at the surface 109
in surface layers 110

annual variation at the surface 1 10-14
in surface layers 114
theory 115
distribution in horizontal and vertical sections

140-49
vertical distribution 117-19
in adjacent seas 129-33
in adjacent seas at higher latitudes 133
mean, for zonal oceanic zones 153
of bottom in the three oceans 149
adiabatic changes 123-6
potential 123, 125
and stability 126-7
Temperature-salinity relationship 202-16
practical significance 203
illustrating water masses 210-14

in Atlantic 202-16
and mixing of water masses 203-10
Thermocline (transition layer)
with physical surface in tropics and subtropics

463
in Atlantic 120
theory 121-22
Topography, absolute, of physical sea surface in
Atlantic Ocean 596
of 100 and 500 decibar isobar surface 517
of 800 and 2000 decibar isobar surface 674,

resp. 686
in convergence region of South Atlantic 657
in the Antarctic convergence zone 670
in Davis Strait and Labrador Sea 668
Trajectory of water movements 342
Transgressions, Atlantic, of Le Danois 659
Transient response of ocean to variable wind

stresses 702-7
Trenches, deep-sea 24-7

and gravity profiles 26-7
Troposphere, oceanic, position and structure
592-602
salinity 166-72
circulation 592-660
Troughs, see Trenches

Turbulence, dissipation of turbence energy 391
and stratification 391
and mixing in the sea 393
statistical theory 393-8
Turbulent friction, see eddy viscosity

Undercurrent, equatorial 604-5
Upwelling phenomena 643-66
theory of Defant, Sverdrup, Hidaka 646-656

Vapour pressure of sea-water 44
Velocity, friction 389
Velocity field 342, 356-370
representation, by means of compass cards 356

by means of stream lines 356
near land 370
divergence of 374-9
Viscosity of sea-water 50
eddy 104

in sea currents 387-391
coefficient 387
Vortex, circular hydrographic, wind eff"ects and

current system 576-9
Vortices, development of 373-4
standing 372

Subject Index

729

Vortices — contd.
stationary in a two layered ocean 465
pulsation of 469-475
Eigen-period of oscillating vortex 474

Water bodies 451

stable stratification between 460^79
Water budget of Earth 231-5
Water interchange between North Sea and

Baltic 526-9
Water masses of oceans 216-17
Water transport in the individual layers of

Atlantic Ocean 688-690
in density current 508-511
in coastal current 511
Wind, effects of, and the current system in a
hydrographic circular vortex 576-9
on mass field and density current 544-555
computation of velocity of equatorial currents

from, data 552-5
relationships between wind and currents
550-2
Wind stress, variable, effect on oceanic currents

702-7
Windstau (piling up of water by wind) 419

W 90' 60° 30° 0° 30° E

E 30" 60" 90" 120" 150° E

i^i

I 150°180°150°1ZO° 90° 60° W

PLJ^■re 1. World-map of ocean depth.
Isobalhs for 1000 and 2000 m and from Ihcre on for each 2000 m-mlcrval. The isohalh are denoted on the maps by 100

Colouring: 0- 1000 m light yellow; 1000 - 4000 m light green; 4000 - 6000 m blue; 0000 m red.

W 60" 30° 0" 30" E

(for example 4000 m = 400).

f X 60 90' E

£• t20°B0° W0°eifl20° W

W 90 60 30 0 30 E

f JO' 60'- 90' KO ISO' E

<iOOO

.. ,. ..J 6000-7000

4000-5000

■Mi 7000-8000

5000 -6000

1^1 >8000

Plate 2. Schemaiically simplified world-map of ocean dcpih. (.S<c : Vol I. Pi 1. p 1 3)
(The depth-iniervals are specified underncalh the map; the lellers and numbers are referred to in the text ; for Altantic Ocean »■*■ p 28. for Indian Ocean *(■<■ pp. i

E 3D 60 30 E

E I2IJ jso ieo m'i2a w

^ go" CO' 30" 0' 30 E

E 30 60 90 120 ISO E

E 150 WO 150 m 90 60" w

W 60 30 0 30 E

Platp 3a. Surface-temperature (°C) of the world oceans for February.
(.9ir Vol I. Pt, I, p, 140 it .vi',/.)

B X 60 90 E

£ }20 m 180 m 120 w

w go" 60° 30" 0° 30° E

E 30 60 90 120 ISO- E

E 150 180 ISO' 120 90 60 W

W 60 30 0' 30" E

Plate 3b. Surface-lemperalure ( C) of the world oceans for August.
(See Vol. 1. PC. I. p. 140 <■( Hcq.)

E lao 150°180°150°li.O'

90° 60° 10° 0° 30° E

E 150°180°150°130" 90° eO" W

60° 30 0 0

Plate 4, Bollom-Icmperalurc ( C) of the world oceans.

(For tempera I urc-inlcrvals indicated by diflTcrent shades, see specifications in the lower righlhand side of ihc map.)

(See Vol, I, Pt. I. p. 149 et scg.)

/~^.

W 90° 60° 30° 0° 30° E

E 30° 60° 90°120°150° E

E 150°ie0°150°120° 90° 60° W

;g 60° 30° 0° 30° E

Plate 5. Average sea-surface salinity {%„) of the world oceans.
(.SVf Vol. I. Pt. T, p. 161 (•/ .?«/.)

W go 60 30 0 30 E

■^^/i.'Y 7tf

IV 60' 30" 0" 30' E

W 60° 30° 0°30° E

Plate 6. Average salinity (%„) in the Atlantic Ocean.

(Picture to tiie left: in 400 m depth; picture to the right: in 1000 m depth.)

(See Vol. I. Pt. 1, p. 179 ct .set/.)

W 9D CO 30 0 30 E

\N 50" CO" 30" O" 30° E

W 60' 30 0 30 E

W 60 30 0 30 E

Plate 7. Average density (o,) in the Atlantic Ocean.

(Picture to the left: in 400 m depth; picture to the right: in 1000 n\ depth.)

(For sca-surfacc sec: Vol. I. Pt. I. Fig. 89, p. 191, for deolh charts p. 192).

E 30° 60° 90° E

"-/' fl/

^_-.

E 30" 60° 90° 120° 150° E

' "" ''"^ ' ""i^^^^^^^

E 150° 180° 160° 120° 90° 60° W

W 60° 30° 0° 30° E

the Northern Hemisphci

Geographical Review

EXCERPT FROM VOL. 52 NO. 4 19 62 pp.624-625

PUBLISHED BY

The American Geographical Society
OF New York

BROADWAY AT 156tli ST., NEW YORK 32, N. Y.

Hudson Laboratories, Columbia University Contribution No. l6l.

PHYSICAL OCEANOGRAPHY. By Albert Defant. Vol. i, xvi and 729 pp.; Vol. 2,
viii and 598 pp.; maps, diagrs., ills., bibliogrs.. indexes. Pergamon Press, New York,
Oxford, London, Paris, 1961. $35.00. 10 x 6^ inches.

The oceans have been described in exciting dramas, which use the violence or desolateness
of the sea to draw out human traits. Except for an occasional storm, tidal wave, or other
such phenomenon, one tends to consider the oceans uninteresting in themselves. Neverthe-
less, by subtle movements resulting from small changes m the properties of water, these
millions of square miles of liquid establish conditions that allow man to exist on earth.
The mechanism behind this forms a part of the general description of oceanic movements
in Defant's two-volume work on physical oceanography. Even marine life is excluded
from this overdue tribute to the seas.

Oceanography has gained much from Professor Defant's past contributions, and it is
fortimate that a man of his stature has written these volumes. His insight and his organiza-
tion of material have resulted in a work that in other hands could have been a shambles
instead of a badly needed coherent description of the state of physical oceanography. He
has considered literature written up to May, 1957, and the vitality of the field is such that
this study can now be regarded as a st.arting reference work.

The first pages of Volume 1 present a description of the oceans — their extent, the dis-
tribution of temperature, salinity, and density, their water budget and conversion into ice
near the poles. The remainder of the volume reviews pertinent physical concepts and applies
them to the general problem of water circulation in all parts ot the oceans. Some welcome
elaborations of hydrodynamic situations are also presented.

Waves and tides form the subject matter ot the second volume. Again, basic physical
ideas are clarified before the author goes into the extensive literature describing the periodic
movements of the sea. The treatment is thorough enough to encompass, for example, the
water transport associated with irrotational surface waves.

Professor Defant wrote this work in German over a period of years. Fortimately, the
Office of Naval Research, United States Navy, was willing to sponsor its translation into
Enghsh, This must have been a formidable undertaking, and the occasional clumsy sentence
constructions are easily forgiven. — T. E. Pochapsky

Wh.

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